# Schur harmonic convexity of Stolarsky extended mean values.

[section]1. Introduction and preliminaries

In literature the importance and applications of means and its inequality to science and technology is explored by eminent researchers see [3]. In [20-28], we studied some results on contra harmonic mean. In [11, 41], authors studied the different properties of the so-called Stolarsky (extended) two parameter mean values, is defined as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1)

Some of the classical two parameter means are special cases.

Here we recall some of the known means which are essential for the paper, the arithmetic mean in weighted form,

[A.sub.p, q](a, b) = pa + qb = A(a, b ; p, q) ;

such that a, b > 0 and p + q = 1, where p and q are the weights.

The Stolarsky means [S.sub.p ; q](a, b) are [C.sup.[infinity]] function on the domain (p, q, a, b), p, q [member of] R, a, b > 0. Obviously, Stolarsky means [S.sub.p ; q](a, b) are symmetric with respect to a, b and p, q. Most of the classical two variable means are special cases of [S.sub.p ; q](a, b), Stolarsky mean. For example:

[E.sub.1, 2](a, b) = a + b/2, is the Arithmetic mean;

[E.sub.0, 0](a, b) = [E.sub.-1,-1](a, b) = [square root of ab], is the Geometric mean;

[E.sub.-1, -2](a, b) = 2ab/a + b, is the Harmonic mean;

[E.sub.0, 1](a, b) = a - b/lna - lnb, is the Logarithmic mean;

[E.sub.1, 1](a, b) = 1/e [([a.sup.a]/[b.sup.b]).sup.1/a - b], is the Identric mean;

[E.sub.r, 2r](a, b) = [([a.sup.r] + [b.sup.r]/2).sup.1/r], is the rth Power mean.

The basic properties of Stolarsky means, as well as their comparison theorems, log-convexities, and inequalities are studied in papers [4, 9, 13, 16, 17, 19, 22, 23, 30-35, 44, 49, 50, 51, 52, 55].

In recent years, the Schur convexity and Schur geometrically convexity of [S.sub.p ; q](a, b) have attracted the attention of a considerable number of mathematicians [5, 6, 18, 36, 38, 40]. Qi [37] first proved that the Stolarsky means [S.sub.p ; q](a, b) are Schur convex on (-[infinity], 0] x (-[infinity], 0] and Schur concave on [0, [infinity]) x [0, [infinity]) with respect to (p, q) for fixed a, b > 0 with a [not equal to] b. Yang [53] improved Qi's result and proved that Stolarsky means [S.sub.p ; q](a, b) are Schur convex with respect to (p, q) for fixed a, b > 0 with a [not equal to] b if and only if p + q < 0 and Schur concave if and only if p + q > 0.

Qi [36] tried to obtain the Schur convexity of [S.sub.p ; q](a, b) with respect to (a, b) for fixed (p, q) and declared an incorrect conclusion. Shi [40] observed that the above conclusion is wrong and obtained a sufficient condition for the Schur convexity of [S.sub.p ; q](a, b) with respect to (a, b). Chu and Zhang [6] improved Shi's results and gave an necessary and sufficient condition. This perfectly solved the Schur convexity of Stolarsky means with respect to (a, b).

For the Schur geometrically convexity, Chu and Zhang [5] proved that Stolarsky means [S.sub.p ; q](a, b) are Schur geometrically convex with respect to (a, b) [member of] (0, [infinity]) x (0, [infinity]) if p + q [greater than or equal to] 0 and Schur geometrically concave if p + q [less than or equal to] 0. Li [18] also investigated the Schur geometrically convexity of generalized exponent mean [I.sub.p](a, b).

The purpose of this paper is to investigate another type of Schur convexity that is the Schur harmonic convexity of Stolarsky means [S.sub.p ; q](a, b), of which the idea is to find the necessary conditions from lemma 2, and then prove these conditions are sufficient.

In [3], the weighted contra harmonic mean is defined on the basis of proportions by,

[C.sub.p, q](a, b) = p[a.sup.2] + q[b.sup.2]/pa + qb = C(a, b ; p, q),

such that a, b > 0 and p + q = 1, where p and q are the weights.

This work motivates us to introduce a new family of Stolarsky's extended type mean values in weighted forms in two and n variables.

For two variables a, b > 0, p, q [member of] R and p, q are the weights, such that p + q = 1, then consider a new mean in the following form:

[N.sub.r, s](a, b ; p, q) = [[[r.sup.2]/C([a.sup.s],[b.sup.s]; p, q) - A([a.sup.s],[b.sup.s]; p, q)/[s.sup.2]C([a.sup.r],[b.sup.r]; p, q) - A([a.sup.r],[b.sup.r]; p, q)].sup.1/s - r],

which is equivalently,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

which is equivalently,

[N.sub.r, s](a, b ; p, q) = [[[r.sup.2]/[s.sup.2] (p[a.sup.r] + q[b.sup.r]/p[a.sup.s] + q[b.sup.s])[([a.sup.s] - [b.sup.s]/[a.sup.r] - [b.sup.r]).sup.2]].sup.1/s - r].

In [50], the authors introduced the homogeneous function with two parameters r and s by,

[H.sub.f](a, b ; s, r) = [[f([a.sup.s], [b.sup.s])/f([a.sup.r], [b.sup.r])].sup.1/s - r],

and studied its monotonicity and deduces some inequalities involving means, where f is a homogeneous function for a and b.

In particular, f = A, is the arithmetic mean of a and b.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2)

Here, [G.sub.A, s](a, b) = [Z.sub.s](a, b) = [Z.sup.1/s]([a.sup.p], [b.sup.p]) = [Z.sub.s]. Z(a, b) = [a.sup.a/a + b] [b.sup.b/a + b] is named power-exponential mean between two positive numbers a and b.

In weighted form,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (3)

In [42], author introduced and studied the various properties and log-convexity results of the class W of weighted two parameter means is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (4)

The above definitions leads to express the mean values [N.sub.r, s](a, b ; p, q) in the following form:

[N.sub.r, s](a, b ; p, q) = [[p[a.sup.r] + q[b.sup.r]/p[a.sup.s] + q[b.sup.s]].sup.1/s - r] [[(r/s ([a.sup.s] - [b.sup.s])/([a.sup.r] - [b.sup.r])).sup.1/s - r].sup.2],

which is equivalently

[N.sub.r, s](a, b ; p, q) = [[f([a.sup.s],[b.sup.s];p, q)/f([a.sup.r],[b.sup.r];p, q)].sup.1/s - r][E.sup.2.sub.r, s] (a, b),

or

[N.sub.r, s](a, b ; p, q) = [H.sub.f = A(a, b ; p, q)](a, b ; s, r)[E.sup.2.sub.r, s](a, b). (5)

Here f = A(a, b ; p, q) is arithmetic mean in weighted form.

The various properties and identities concerning to [N.sub.r, s](a, b ; p, q) are also listed. The laborious calculations gives the following different cases of the mean value [N.sub.r, s](a, b ; p, q).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (6)

[section]2. Definition and properties

Schur convexity was introduced by Schur in 1923 [24], and it has many important applications in analytic inequalities [2, 12, 54], linear regression [43], graphs and matrices [8], combinatorial optimization [15], information theoretic topics [10], Gamma functions [25], stochastic orderings [39], reliability [14], and other related fields. Recently, Anderson [1] discussed an attractive class of inequalities, which arise from the notation of harmonic convexity. For convenience of readers, we recall some definitions as follows.

We recall the definitions which are essential to develop this paper.

Definition 2. [24, 46] Let x = ([x.sub.1], [x.sub.2], ..., [x.sub.n]) and y = ([y.sub.1], [y.sub.2], ..., [y.sub.n]) [member of] [R.sup.n],

1. x is majorized by y, (in symbol x < y). If [[summation].sup.k.sub.i = 1][x.sub.[i]] [less than or equal to] [[summation].sup.k.sub.i = 1][y.sub.[i]] and [[summation].sup.n.sub.i = 1][x.sub.[i]] [less than or equal to] [[summation].sup.n.sub.i = 1][y.sub.[i]],

where [x.sub.[1]] [greater than or equal to] ... [greater than or equal to] [x.sub.[n]] and [y.sub.[1]] [greater than or equal to] ... [greater than or equal to] [y.sub.[n]] are rearrangements of x and y in descending order.

2. x [greater than or equal to] y means [x.sub.i] [greater than or equal to] [y.sub.i] for all i = 1, 2, ..., n. Let [OMEGA] [member of] [R.sup.n](n [greater than or equal to] 2). The function [phi] : [OMEGA] [right arrow] R is said to be decreasing if and only if-[phi] is increasing.

3. [OMEGA] [subset or equal to] [R.sup.n] is called a convex set if ([alpha][x.sub.1] + [beta][y.sub.1], ..., [alpha][x.sub.n] + [beta][y.sub.n]) for every x and y [member of] [OMEGA] where [alpha], [beta] [member of] [0, 1] with [alpha] + [beta] = 1.

4. Let [OMEGA] [subset or equal to] [R.sup.n] the function [phi] : [OMEGA] [right arrow] R be said to be a schur convex function on [OMEGA] if x [less than or equal to] y on [OMEGA] implies [phi](x) [less than or equal to] [phi](y). [phi] is said to be a schur concave function on [OMEGA] if and only if -[phi] is schur convex.

Definition 2.2. [55] Let x = ([x.sub.1], [x.sub.2], ..., [x.sub.n]) and y = ([y.sub.1], [y.sub.2], ..., [y.sub.n]) [member of] [R.sup.n.sub.+], [OMEGA] [subset or equal to] [R.sup.n] is called Harmonically convex set if ([x.sup.[alpha].sub.1][y.sup.[beta].sub.1], ..., [x.sup.[alpha].sub.1] [y.sup.[beta].sub.1]) [member of] [OMEGA] for all x and y [member of] [OMEGA], where [alpha], [beta] [member of] [0, 1] with [alpha] + [beta] = 1.

Let [OMEGA] [subset or equal to] [R.sup.n.sub.+], the function [phi] : [OMEGA] [right arrow] [R.sub.+] is said to be schur Harmonically convex function on [OMEGA], if (ln[x.sub.1], ..., l[n.sub.n]) < (ln[y.sub.1], ..., ln[y.sub.n]) on [OMEGA] implies [phi](x) [less than or equal to] [phi](y). Then [phi] is said to be a schur Harmonically concave function on [OMEGA] if and only if-[phi] is schur Harmonically convex.

Definition 2.3. [24, 46] Let [OMEGA] [subset or equal to] [R.sup.n] is called symmetric set if x [member of] [OMEGA] implies Px [member of] [OMEGA] for every n x n permutation matrix P, the function [phi] : [OMEGA] [right arrow] R is called symmetric if for every permutation matrix P, [phi](Px) = [phi](x) for all x [member of] [OMEGA].

Definition 2.4. [24, 46] Let [OMEGA] [subset or equal to] [R.sup.n], [phi] : [OMEGA] [right arrow] R is called symmetric and convex function. Then [phi] is schur convex on [OMEGA].

Lemma 2.1. [55] Let [OMEGA] [subset or equal to] [R.sup.n] be symmetric with non empty interior Harmonically convex set and let [phi] : [OMEGA] [right arrow] [R.sub.+] be continuous on [OMEGA] and differentiate in [[OMEGA].sup.0]. If [phi] is symmetric on [OMEGA] and

([x.sub.1] - [x.sub.2])([x.sup.2.sub.1][partial derivative][phi]/[partial derivative][x.sub.1] - [x.sup.2.sub.2][partial derivative][phi]/[partial derivative][x.sub.2]) [greater than or equal to] 0 ([less than or equal to] 0), (7)

holds for any x = ([x.sub.1], [x.sub.2], ..., [x.sub.n]) [member of] [[OMEGA].sup.0], then [phi] is a schur- Harmonically convex (Schur-Harmonically) concave function.

Lemma 2.2. [40] Let a [less than or equal to] b, u(t) = ta + (1 - t)b, v(t) = tb + (1 - t)a, 1/2 [less than or equal to] [t.sub.2] [less than or equal to] [t.sub.1] [less than or equal to] 1, 0 [less than or equal to] [t.sub.1] [less than or equal to] [t.sub.2] [less than or equal to] 1/2, then

(a + b)/2 < (u([t.sub.2]), v([t.sub.2])) < (u([t.sub.1]), v([t.sub.1])). (8)

[section]3. Main results

In this section, we shall prove some the lemmas required for proving main theorem.

Lemma 3.1. Stolarsky's extended family type means [N.sub.p, q](a, b ; r, s) are schur-Harmonic convex or Schur-Harmonic concave with respect to (a, b) [member of] (0, [infinity]) x (0, [infinity]) if and only if g(t) [greater than or equal to] 0 or g(t) [less than or equal to] 0 for all t > 0, where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (9)

and

A = p + q + 1, B = p - q + 1, C = p - q - 1, D = p + q, E = p - q. (10)

Proof. Let Stolarsky's extended family type mean N = [N.sub.p, q](a, b ; r, s) defined for pq(p - q) [not equal to] 0 as

N = [N.sub.p, q](a, b ; r, s) = [[q.sup.2]/[p.sup.2](r[a.sup.p] + s[b.sup.p]/r[a.sup.q] + s[b.sup.q])[([a.sup.p] - [b.sup.p]/[a.sup.q] - [b.sup.q]).sup.2]].sup.1/p - q. (11)

Let r = s = 1/2, take log on both sides and differentiate partially with respect to a and multiply by [a.sup.2], gives

[a.sup.2][partial derivative]N/[partial derivative]a = N/p - q [q[a.sup.q + 1]/[a.sup.q] + [b.sup.q] - p[a.sup.p + 1]/[a.sup.p] + [b.sup.p] + 2 p[a.sup.p + 1]/[a.sup.p] - [b.sup.p] - 2q[a.sup.q + 1]/[a.sup.q] - [b.sup.q]]. (12)

Similarly,

[b.sup.2][partial derivative]N/[partial derivative]a = N/p - q[q[b.sup.q + 1]/[a.sup.q] + [b.sup.q] - p[b.sup.p + 1]/[a.sup.p] + [b.sup.p] + 2 p[b.sup.p + 1]/[a.sup.p] - [b.sup.p] - 2 q[b.sup.q + 1]/[a.sup.q] - [b.sup.q]], (13)

then,

(a - b)([a.sup.2][partial derivative]G/[partial derivative]a - [b.sup.2][partial derivative]G/[partial derivative]b) = (a - b)N/p - q[[DELTA]], (14)

where,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Substituting [square root of a/b] = t and using sinh x = 1/2([e.sup.x] - [e.sup.-x]), cosh x = 1/2([e.sup.x] + [e.sup.- x]), we have

[DELTA] = [square root of ab][q sinh(q + 1)t/cosh qt - p sinh(p + 1)t/cosh pt + 2p cosh(p + 1)t/sinh pt - 2q cosh(q + 1)t/sinh qt].

Using the product into sum formula for hyperbolic functions leads to: For pq(p - q) [not equal to] 0

(a - b)([a.sup.2] [partial derivative]G/[partial derivative]a - [b.sup.2][partial derivative]G/[partial derivative]b) = (pq)N(a - b)[square root of ab]/2 sinh pt sinh qt cosh pt cosh qt[[g.sub.p, q](t)], (15)

where,

[g.sub.p, q](t) = [(p - q) sinh At{cosh(p + q)t + 3cosh (p - q)t}/pq(p - q)] - [(p sinh Bt + q sinh Ct){3cosh(p + q)t + cosh(p - q)t}/pq(p - q)]. (16)

In case of p [not equal to] q = 0. Since [N.sub.p, q] [member of] [C.sup.[infinity]], we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

Thus

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Likewise for q [not equal to] p = 0,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

For q = p [not equal to] 0,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

For q = p = 0,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

By summarizing all cases above yield

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (17)

Since (a - b)([a.sup.2][partial derivative]N/[partial derivative]a - [b.sup.2][partial derivative]N/[partial derivative]b) is symmetric with respect to a and b, without loss of generality we assume a > b, then t = ln[square root of a/b] > 0. It is easy to verify that N(a - b)[square root of ab]/2 > 0, p/sinh pt, q/sinh qt > 0, if pq [not equal to] 0 for t > 0. Thus by lemma 2 Stolarsky means [S.sub.p, q](a, b) are Schur harmonic convex (Schur harmonic concave) with respect to (a, b) [member of] (0, [infinity]) x (0, [infinity]), if and only if (a - b)([a.sup.2][partial derivative]N/[partial derivative]a - [b.sup.2][partial derivative]N/[partial derivative]b)([greater than or equal to])([less than or equal to])0, if and only if g(t) = [g.sub.p, q](t)([greater than or equal to]) ([less than or equal to]) 0 for all t > 0. This completes the proof of lemma 3.1.

Lemma 3.2. The function g(t) = [g.sub.p, q](t) defined by (3.1) and g'(t) = [partial derivative][g.sub.p, q](t)/[partial derivative]t both are symmetric with respect to p and q, and both continuous with respect to p and q on R x R.

Proof. It is easy to check that [g.sub.p, q](t) and [partial derivative][g.sub.p, q](t)/[partial derivative]t are symmetric with respect to p and q, then [partial derivative][g.sub.p, q](t)/[partial derivative]t = [partial derivative][g.sub.q, p](t)/[partial derivative]t.

By lemma 3.1, we note that g(t) = [g.sub.p, q](t) is continuous with respect to p and q on R x R. Finally, we prove that g'(t) = [partial derivative][g.sub.p, q](t)/[partial derivative]t is also continuous with respect to p and q on R x R. A simple calculations yield, Case i: For pq(p - q) [not equal to] 0,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Case ii: For q = 0, p [not equal to] 0,

g'(t) = [partial derivative][g.sub.p, q](t)/[partial derivative]t = 2p cosh(1 + 2p)t + 2pt(1 + 2p) sinh (2p + 1)t - 6p cosht/-[p.sup.2] - 6pt sinht - 2(2p + 1) cosh(2p + 1)t - (2p - 1) cosh(2p - 1)t/-[p.sup.2].

Case iii: For p = 0, q [not equal to] 0,

g'(t) = [partial derivative][g.sub.p, q](t)/[partial derivative]t = 2q cosh(1 + 2q)t + 2qt(1 + 2q)sinh(2q + 1)t - 6q cosh t/-[p.sup.2] - 6qt sinht - 2(2q + 1)cosh(2q + 1)t - (2q - 1)cosh(2q - 1)t/-[p.sup.2].

Case iv: For p = q [not equal to] 0,

g'(t) = [partial derivative][g.sub.p, q](t)/[partial derivative]t = (1 + 2q)(3 + cosh 2qt)cosh(2q + 1)t - 6q (sinh 2qt)(2qt cosh t + sinh t)/[q.sup.2] - (1 + 3 cosh 2qt)((1 + 2q)cosh t + 2qt sinh t) + 2q(sinh 2qt)sinh(2q + 1)t/[q.sup.2].

Case v: For p = q = 0,

g'(t) = [partial derivative][g.sub.p, q](t)/[partial derivative]t = 4[t.sup.2]cosh t + 8t sinh t.

It is obvious that [partial derivative][g.sub.p, q](t)/[partial derivative]t is continuous with respect to p and q on R x R, again in view of

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

These arguments leads that the above five cases are continuous for all values of p and q. This completes the proof of lemma 3.2.

Lemma 3.3. [lim.sub.t [right arrow] 0, t > 0][t.sup.-3]g(t) = -4/3(p + q - 3). Proof. It is easy to check that first and second derivatives of g(t) = 0, at t = 0. In the case of pq(p - q) [not equal to] 0. Applying L-Hospital's rule (three times) yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Similarly, for p = 0, q [not equal to] 0,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];

for q = 0, p [not equal to] 0,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];

for p = q [not equal to] 0,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];

for p = q = 0,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

This completes the proof.

The proof of our main result stated below follows from the above lemmas:

Theorem 3.1. For fixed (p, q) [member of] R x R,

(1) Stolarsky's extended family type mean means [N.sub.r, s](a, b ; p, q) are Schur harmonic convex with respect to (a, b) if p + q - 3 [greater than or equal to] 0.

(2) Stolarsky's extended family type mean means [N.sub.r, s](a, b ; p, q) are Schur harmonic concave if p + q - 3 [less than or equal to] 0.

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[26] K. M. Nagaraja, V. Lokesha and S. Padmanabhan, A simple proof on strengthening and extension of inequalities, Advn. Stud. Contemp. Math., 17(2008), No. 1, 97-103.

[27] K. M. Nagaraja and P. Siva Kota Reddy, [alpha]-Centroidal mean and its dual, Proceedings of the Jangjeon Math. Soc, 15(2012), No. 2, 163-170.

[28] N. Kumar, K. M. Nagaraja, A. Bayad and M. Saraj, New means and its properties, Proceedings of the Jangjeon Math. Soc, 14(2010), No. 3, 243-254.

[29] C. P. Niculescu, Convexity according to the geometric mean, Math. Inequal. Appl., 3(2000), No. 2, 155-167.

[30] E. Neuman and J. Sdor, Inequalities involving Stolarsky and Gini means, Math. Pannon., 14(2003), No. 1, 29-44.

[31] E. Neuman and Zs. Pes, On comparison of Stolarsky and Gini means, J. Math. Anal. Appl., 278(2003), No. 2, 274-284.

[32] Zs. Pes, Inequalities for differences of powers, J. Math. Anal. Appl., 131(1988), 271-281.

[33] Zs. Pes, Comparison of two variable homogeneous means, General Inequal. 6. Proc. 6-th Internat. Conf. Math. Res. Inst. Oberwolfach, Birkhser Verlag Basel, (1992), 59-69.

[34] F. QI, Logarithmically convexities of the Extended Mean values, Proc. Amer. Math. Soc, 130(2002), No. 6, 1787-1796.

[35] F. QI, The Extended mean values: definition, properties, monotonicities, comparison, con-vexities, generalizations, and applications, RGMIA Res. Rep. Coll., 2(2002), No. 1, art. 6, available online at http: //rgmia.vu.edu.au/v2n5.html.

[36] F. QI, J. Sdor and S. S. Dragomir, Notes on the Schur-convexity of the extended mean values, Taiwanese J. Math., 9(2005), No. 3, 411-420.

[37] F. QI, A note on Schur-convexity of extended mean values, Rocky Mountain J. Math., 35(2005), No. 5, 1787-1793.

[38] J. Sdor, The Schur-convexity of Stolarsky and Gini means, Banach J. Math. Anal., 1(2007), No. 2, 212-215.

[39] M. Shaked, J. G. Shanthikumar and Y. L. Tong, Parametric Schur convexity and arrangement monotonicity properties of partial sums, J. Multivariate Anal., 53(1995), No. 2, 293-310.

[40] H. N. Shi, S. H. Wu and F. QI, An alternative note on the Schur-convexity of the extended mean values, Math. Inequal. AppL, 9(2006), No. 2, 219-224.

[41] Slavko Simic, An extension of Stolarsky means, Novi Sad J. Math., 38(2008), No. 3, 81-89.

[42] Slavko Simic, On weighted Stolarsky means, Sarajevo Journal of Mathematics, 7(2011), No. 19, 3-9.

[43] C. Stepniak, Stochastic ordering and Schur-convex functions in comparison of linear experiments, Metrika, 36(1989), No. 5, 291-298.

[44] K. B. Stolarsky, Generalizations of the Logarithmic Mean, Math. Mag., 48(1975), 87-92.

[45] K. B. Stolarsky, The power and generalized Logarithmic Means, Amer. Math. Monthly., 87(1980), 545-548.

[46] B. Y. Wang, Foundations of majorization inequalities, Beijing Normal Univ. Press, Beijing, China, 1990. (Chinese)

[47] W. F. Xia, The Schur harmonic convexity of Lehmer means, Int. Math. Forum, 41(2009), No. 4, 2009-2015.

[48] W. F. Xia and Y. M. Chu, Schur-convexity for a class of symmetric functions and its applications, J. Inequal. AppL, Vol. 2009, Art. ID 493759, 15 pages.

[49] ZH. H. Yang, Simple discriminances of convexity of homogeneous functions and applications, 4(2004), No. 7, 14-19.

[50] ZH. H. Yang, On the homogeneous functions with two parameters and its monotonicity, J. Inequal. Pure App Math., 6(2005), No. 4, art. 101, available online at http: //jipam.vu.edu. au/images/155-05. pdf.

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[52] ZH. H. Yang, On the monotonicity and log-convexity of a four-parameter homogeneous mean, J. Inequal. AppL, Vol. 2008, Art. ID 149286, 2008, available online at http: //www. hindawi. com/GetArticle. aspx?doi = 10.1155/2008/149286.

[53] ZH. H. Yang, Necessary and sufficient conditions for Schur convexity of Stolarsky and Gini means. Submitted.

[54] X. M. Zhang, Schur-convex functions and isoperimetric inequalities, Proc. Amer. Math. Soc, 126(1998), No. 2, 461-470.

[55] X. M. Zhang, Geometrically convex functions, Hefei, An hui University Press, 2004.

K. M. Nagaraja ([dagger]) and Sudhir Kumar Sahu ([double dagger])

([dagger]) Department of Mathematics, JSS Academy of Technical Education, Uttarahalli-Kengeri Main Road, Bangalore-60, Karanataka, India

([double dagger]) Post Graduate, Department of Statistics, Sambalpur University, Sambalpur, Orissa-768019, India

E-mails: nagkmn@gmail.com drsudhir1972@gmail.com

In literature the importance and applications of means and its inequality to science and technology is explored by eminent researchers see [3]. In [20-28], we studied some results on contra harmonic mean. In [11, 41], authors studied the different properties of the so-called Stolarsky (extended) two parameter mean values, is defined as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1)

Some of the classical two parameter means are special cases.

Here we recall some of the known means which are essential for the paper, the arithmetic mean in weighted form,

[A.sub.p, q](a, b) = pa + qb = A(a, b ; p, q) ;

such that a, b > 0 and p + q = 1, where p and q are the weights.

The Stolarsky means [S.sub.p ; q](a, b) are [C.sup.[infinity]] function on the domain (p, q, a, b), p, q [member of] R, a, b > 0. Obviously, Stolarsky means [S.sub.p ; q](a, b) are symmetric with respect to a, b and p, q. Most of the classical two variable means are special cases of [S.sub.p ; q](a, b), Stolarsky mean. For example:

[E.sub.1, 2](a, b) = a + b/2, is the Arithmetic mean;

[E.sub.0, 0](a, b) = [E.sub.-1,-1](a, b) = [square root of ab], is the Geometric mean;

[E.sub.-1, -2](a, b) = 2ab/a + b, is the Harmonic mean;

[E.sub.0, 1](a, b) = a - b/lna - lnb, is the Logarithmic mean;

[E.sub.1, 1](a, b) = 1/e [([a.sup.a]/[b.sup.b]).sup.1/a - b], is the Identric mean;

[E.sub.r, 2r](a, b) = [([a.sup.r] + [b.sup.r]/2).sup.1/r], is the rth Power mean.

The basic properties of Stolarsky means, as well as their comparison theorems, log-convexities, and inequalities are studied in papers [4, 9, 13, 16, 17, 19, 22, 23, 30-35, 44, 49, 50, 51, 52, 55].

In recent years, the Schur convexity and Schur geometrically convexity of [S.sub.p ; q](a, b) have attracted the attention of a considerable number of mathematicians [5, 6, 18, 36, 38, 40]. Qi [37] first proved that the Stolarsky means [S.sub.p ; q](a, b) are Schur convex on (-[infinity], 0] x (-[infinity], 0] and Schur concave on [0, [infinity]) x [0, [infinity]) with respect to (p, q) for fixed a, b > 0 with a [not equal to] b. Yang [53] improved Qi's result and proved that Stolarsky means [S.sub.p ; q](a, b) are Schur convex with respect to (p, q) for fixed a, b > 0 with a [not equal to] b if and only if p + q < 0 and Schur concave if and only if p + q > 0.

Qi [36] tried to obtain the Schur convexity of [S.sub.p ; q](a, b) with respect to (a, b) for fixed (p, q) and declared an incorrect conclusion. Shi [40] observed that the above conclusion is wrong and obtained a sufficient condition for the Schur convexity of [S.sub.p ; q](a, b) with respect to (a, b). Chu and Zhang [6] improved Shi's results and gave an necessary and sufficient condition. This perfectly solved the Schur convexity of Stolarsky means with respect to (a, b).

For the Schur geometrically convexity, Chu and Zhang [5] proved that Stolarsky means [S.sub.p ; q](a, b) are Schur geometrically convex with respect to (a, b) [member of] (0, [infinity]) x (0, [infinity]) if p + q [greater than or equal to] 0 and Schur geometrically concave if p + q [less than or equal to] 0. Li [18] also investigated the Schur geometrically convexity of generalized exponent mean [I.sub.p](a, b).

The purpose of this paper is to investigate another type of Schur convexity that is the Schur harmonic convexity of Stolarsky means [S.sub.p ; q](a, b), of which the idea is to find the necessary conditions from lemma 2, and then prove these conditions are sufficient.

In [3], the weighted contra harmonic mean is defined on the basis of proportions by,

[C.sub.p, q](a, b) = p[a.sup.2] + q[b.sup.2]/pa + qb = C(a, b ; p, q),

such that a, b > 0 and p + q = 1, where p and q are the weights.

This work motivates us to introduce a new family of Stolarsky's extended type mean values in weighted forms in two and n variables.

For two variables a, b > 0, p, q [member of] R and p, q are the weights, such that p + q = 1, then consider a new mean in the following form:

[N.sub.r, s](a, b ; p, q) = [[[r.sup.2]/C([a.sup.s],[b.sup.s]; p, q) - A([a.sup.s],[b.sup.s]; p, q)/[s.sup.2]C([a.sup.r],[b.sup.r]; p, q) - A([a.sup.r],[b.sup.r]; p, q)].sup.1/s - r],

which is equivalently,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

which is equivalently,

[N.sub.r, s](a, b ; p, q) = [[[r.sup.2]/[s.sup.2] (p[a.sup.r] + q[b.sup.r]/p[a.sup.s] + q[b.sup.s])[([a.sup.s] - [b.sup.s]/[a.sup.r] - [b.sup.r]).sup.2]].sup.1/s - r].

In [50], the authors introduced the homogeneous function with two parameters r and s by,

[H.sub.f](a, b ; s, r) = [[f([a.sup.s], [b.sup.s])/f([a.sup.r], [b.sup.r])].sup.1/s - r],

and studied its monotonicity and deduces some inequalities involving means, where f is a homogeneous function for a and b.

In particular, f = A, is the arithmetic mean of a and b.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2)

Here, [G.sub.A, s](a, b) = [Z.sub.s](a, b) = [Z.sup.1/s]([a.sup.p], [b.sup.p]) = [Z.sub.s]. Z(a, b) = [a.sup.a/a + b] [b.sup.b/a + b] is named power-exponential mean between two positive numbers a and b.

In weighted form,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (3)

In [42], author introduced and studied the various properties and log-convexity results of the class W of weighted two parameter means is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (4)

The above definitions leads to express the mean values [N.sub.r, s](a, b ; p, q) in the following form:

[N.sub.r, s](a, b ; p, q) = [[p[a.sup.r] + q[b.sup.r]/p[a.sup.s] + q[b.sup.s]].sup.1/s - r] [[(r/s ([a.sup.s] - [b.sup.s])/([a.sup.r] - [b.sup.r])).sup.1/s - r].sup.2],

which is equivalently

[N.sub.r, s](a, b ; p, q) = [[f([a.sup.s],[b.sup.s];p, q)/f([a.sup.r],[b.sup.r];p, q)].sup.1/s - r][E.sup.2.sub.r, s] (a, b),

or

[N.sub.r, s](a, b ; p, q) = [H.sub.f = A(a, b ; p, q)](a, b ; s, r)[E.sup.2.sub.r, s](a, b). (5)

Here f = A(a, b ; p, q) is arithmetic mean in weighted form.

The various properties and identities concerning to [N.sub.r, s](a, b ; p, q) are also listed. The laborious calculations gives the following different cases of the mean value [N.sub.r, s](a, b ; p, q).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (6)

[section]2. Definition and properties

Schur convexity was introduced by Schur in 1923 [24], and it has many important applications in analytic inequalities [2, 12, 54], linear regression [43], graphs and matrices [8], combinatorial optimization [15], information theoretic topics [10], Gamma functions [25], stochastic orderings [39], reliability [14], and other related fields. Recently, Anderson [1] discussed an attractive class of inequalities, which arise from the notation of harmonic convexity. For convenience of readers, we recall some definitions as follows.

We recall the definitions which are essential to develop this paper.

Definition 2. [24, 46] Let x = ([x.sub.1], [x.sub.2], ..., [x.sub.n]) and y = ([y.sub.1], [y.sub.2], ..., [y.sub.n]) [member of] [R.sup.n],

1. x is majorized by y, (in symbol x < y). If [[summation].sup.k.sub.i = 1][x.sub.[i]] [less than or equal to] [[summation].sup.k.sub.i = 1][y.sub.[i]] and [[summation].sup.n.sub.i = 1][x.sub.[i]] [less than or equal to] [[summation].sup.n.sub.i = 1][y.sub.[i]],

where [x.sub.[1]] [greater than or equal to] ... [greater than or equal to] [x.sub.[n]] and [y.sub.[1]] [greater than or equal to] ... [greater than or equal to] [y.sub.[n]] are rearrangements of x and y in descending order.

2. x [greater than or equal to] y means [x.sub.i] [greater than or equal to] [y.sub.i] for all i = 1, 2, ..., n. Let [OMEGA] [member of] [R.sup.n](n [greater than or equal to] 2). The function [phi] : [OMEGA] [right arrow] R is said to be decreasing if and only if-[phi] is increasing.

3. [OMEGA] [subset or equal to] [R.sup.n] is called a convex set if ([alpha][x.sub.1] + [beta][y.sub.1], ..., [alpha][x.sub.n] + [beta][y.sub.n]) for every x and y [member of] [OMEGA] where [alpha], [beta] [member of] [0, 1] with [alpha] + [beta] = 1.

4. Let [OMEGA] [subset or equal to] [R.sup.n] the function [phi] : [OMEGA] [right arrow] R be said to be a schur convex function on [OMEGA] if x [less than or equal to] y on [OMEGA] implies [phi](x) [less than or equal to] [phi](y). [phi] is said to be a schur concave function on [OMEGA] if and only if -[phi] is schur convex.

Definition 2.2. [55] Let x = ([x.sub.1], [x.sub.2], ..., [x.sub.n]) and y = ([y.sub.1], [y.sub.2], ..., [y.sub.n]) [member of] [R.sup.n.sub.+], [OMEGA] [subset or equal to] [R.sup.n] is called Harmonically convex set if ([x.sup.[alpha].sub.1][y.sup.[beta].sub.1], ..., [x.sup.[alpha].sub.1] [y.sup.[beta].sub.1]) [member of] [OMEGA] for all x and y [member of] [OMEGA], where [alpha], [beta] [member of] [0, 1] with [alpha] + [beta] = 1.

Let [OMEGA] [subset or equal to] [R.sup.n.sub.+], the function [phi] : [OMEGA] [right arrow] [R.sub.+] is said to be schur Harmonically convex function on [OMEGA], if (ln[x.sub.1], ..., l[n.sub.n]) < (ln[y.sub.1], ..., ln[y.sub.n]) on [OMEGA] implies [phi](x) [less than or equal to] [phi](y). Then [phi] is said to be a schur Harmonically concave function on [OMEGA] if and only if-[phi] is schur Harmonically convex.

Definition 2.3. [24, 46] Let [OMEGA] [subset or equal to] [R.sup.n] is called symmetric set if x [member of] [OMEGA] implies Px [member of] [OMEGA] for every n x n permutation matrix P, the function [phi] : [OMEGA] [right arrow] R is called symmetric if for every permutation matrix P, [phi](Px) = [phi](x) for all x [member of] [OMEGA].

Definition 2.4. [24, 46] Let [OMEGA] [subset or equal to] [R.sup.n], [phi] : [OMEGA] [right arrow] R is called symmetric and convex function. Then [phi] is schur convex on [OMEGA].

Lemma 2.1. [55] Let [OMEGA] [subset or equal to] [R.sup.n] be symmetric with non empty interior Harmonically convex set and let [phi] : [OMEGA] [right arrow] [R.sub.+] be continuous on [OMEGA] and differentiate in [[OMEGA].sup.0]. If [phi] is symmetric on [OMEGA] and

([x.sub.1] - [x.sub.2])([x.sup.2.sub.1][partial derivative][phi]/[partial derivative][x.sub.1] - [x.sup.2.sub.2][partial derivative][phi]/[partial derivative][x.sub.2]) [greater than or equal to] 0 ([less than or equal to] 0), (7)

holds for any x = ([x.sub.1], [x.sub.2], ..., [x.sub.n]) [member of] [[OMEGA].sup.0], then [phi] is a schur- Harmonically convex (Schur-Harmonically) concave function.

Lemma 2.2. [40] Let a [less than or equal to] b, u(t) = ta + (1 - t)b, v(t) = tb + (1 - t)a, 1/2 [less than or equal to] [t.sub.2] [less than or equal to] [t.sub.1] [less than or equal to] 1, 0 [less than or equal to] [t.sub.1] [less than or equal to] [t.sub.2] [less than or equal to] 1/2, then

(a + b)/2 < (u([t.sub.2]), v([t.sub.2])) < (u([t.sub.1]), v([t.sub.1])). (8)

[section]3. Main results

In this section, we shall prove some the lemmas required for proving main theorem.

Lemma 3.1. Stolarsky's extended family type means [N.sub.p, q](a, b ; r, s) are schur-Harmonic convex or Schur-Harmonic concave with respect to (a, b) [member of] (0, [infinity]) x (0, [infinity]) if and only if g(t) [greater than or equal to] 0 or g(t) [less than or equal to] 0 for all t > 0, where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (9)

and

A = p + q + 1, B = p - q + 1, C = p - q - 1, D = p + q, E = p - q. (10)

Proof. Let Stolarsky's extended family type mean N = [N.sub.p, q](a, b ; r, s) defined for pq(p - q) [not equal to] 0 as

N = [N.sub.p, q](a, b ; r, s) = [[q.sup.2]/[p.sup.2](r[a.sup.p] + s[b.sup.p]/r[a.sup.q] + s[b.sup.q])[([a.sup.p] - [b.sup.p]/[a.sup.q] - [b.sup.q]).sup.2]].sup.1/p - q. (11)

Let r = s = 1/2, take log on both sides and differentiate partially with respect to a and multiply by [a.sup.2], gives

[a.sup.2][partial derivative]N/[partial derivative]a = N/p - q [q[a.sup.q + 1]/[a.sup.q] + [b.sup.q] - p[a.sup.p + 1]/[a.sup.p] + [b.sup.p] + 2 p[a.sup.p + 1]/[a.sup.p] - [b.sup.p] - 2q[a.sup.q + 1]/[a.sup.q] - [b.sup.q]]. (12)

Similarly,

[b.sup.2][partial derivative]N/[partial derivative]a = N/p - q[q[b.sup.q + 1]/[a.sup.q] + [b.sup.q] - p[b.sup.p + 1]/[a.sup.p] + [b.sup.p] + 2 p[b.sup.p + 1]/[a.sup.p] - [b.sup.p] - 2 q[b.sup.q + 1]/[a.sup.q] - [b.sup.q]], (13)

then,

(a - b)([a.sup.2][partial derivative]G/[partial derivative]a - [b.sup.2][partial derivative]G/[partial derivative]b) = (a - b)N/p - q[[DELTA]], (14)

where,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Substituting [square root of a/b] = t and using sinh x = 1/2([e.sup.x] - [e.sup.-x]), cosh x = 1/2([e.sup.x] + [e.sup.- x]), we have

[DELTA] = [square root of ab][q sinh(q + 1)t/cosh qt - p sinh(p + 1)t/cosh pt + 2p cosh(p + 1)t/sinh pt - 2q cosh(q + 1)t/sinh qt].

Using the product into sum formula for hyperbolic functions leads to: For pq(p - q) [not equal to] 0

(a - b)([a.sup.2] [partial derivative]G/[partial derivative]a - [b.sup.2][partial derivative]G/[partial derivative]b) = (pq)N(a - b)[square root of ab]/2 sinh pt sinh qt cosh pt cosh qt[[g.sub.p, q](t)], (15)

where,

[g.sub.p, q](t) = [(p - q) sinh At{cosh(p + q)t + 3cosh (p - q)t}/pq(p - q)] - [(p sinh Bt + q sinh Ct){3cosh(p + q)t + cosh(p - q)t}/pq(p - q)]. (16)

In case of p [not equal to] q = 0. Since [N.sub.p, q] [member of] [C.sup.[infinity]], we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

Thus

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Likewise for q [not equal to] p = 0,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

For q = p [not equal to] 0,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

For q = p = 0,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

By summarizing all cases above yield

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (17)

Since (a - b)([a.sup.2][partial derivative]N/[partial derivative]a - [b.sup.2][partial derivative]N/[partial derivative]b) is symmetric with respect to a and b, without loss of generality we assume a > b, then t = ln[square root of a/b] > 0. It is easy to verify that N(a - b)[square root of ab]/2 > 0, p/sinh pt, q/sinh qt > 0, if pq [not equal to] 0 for t > 0. Thus by lemma 2 Stolarsky means [S.sub.p, q](a, b) are Schur harmonic convex (Schur harmonic concave) with respect to (a, b) [member of] (0, [infinity]) x (0, [infinity]), if and only if (a - b)([a.sup.2][partial derivative]N/[partial derivative]a - [b.sup.2][partial derivative]N/[partial derivative]b)([greater than or equal to])([less than or equal to])0, if and only if g(t) = [g.sub.p, q](t)([greater than or equal to]) ([less than or equal to]) 0 for all t > 0. This completes the proof of lemma 3.1.

Lemma 3.2. The function g(t) = [g.sub.p, q](t) defined by (3.1) and g'(t) = [partial derivative][g.sub.p, q](t)/[partial derivative]t both are symmetric with respect to p and q, and both continuous with respect to p and q on R x R.

Proof. It is easy to check that [g.sub.p, q](t) and [partial derivative][g.sub.p, q](t)/[partial derivative]t are symmetric with respect to p and q, then [partial derivative][g.sub.p, q](t)/[partial derivative]t = [partial derivative][g.sub.q, p](t)/[partial derivative]t.

By lemma 3.1, we note that g(t) = [g.sub.p, q](t) is continuous with respect to p and q on R x R. Finally, we prove that g'(t) = [partial derivative][g.sub.p, q](t)/[partial derivative]t is also continuous with respect to p and q on R x R. A simple calculations yield, Case i: For pq(p - q) [not equal to] 0,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Case ii: For q = 0, p [not equal to] 0,

g'(t) = [partial derivative][g.sub.p, q](t)/[partial derivative]t = 2p cosh(1 + 2p)t + 2pt(1 + 2p) sinh (2p + 1)t - 6p cosht/-[p.sup.2] - 6pt sinht - 2(2p + 1) cosh(2p + 1)t - (2p - 1) cosh(2p - 1)t/-[p.sup.2].

Case iii: For p = 0, q [not equal to] 0,

g'(t) = [partial derivative][g.sub.p, q](t)/[partial derivative]t = 2q cosh(1 + 2q)t + 2qt(1 + 2q)sinh(2q + 1)t - 6q cosh t/-[p.sup.2] - 6qt sinht - 2(2q + 1)cosh(2q + 1)t - (2q - 1)cosh(2q - 1)t/-[p.sup.2].

Case iv: For p = q [not equal to] 0,

g'(t) = [partial derivative][g.sub.p, q](t)/[partial derivative]t = (1 + 2q)(3 + cosh 2qt)cosh(2q + 1)t - 6q (sinh 2qt)(2qt cosh t + sinh t)/[q.sup.2] - (1 + 3 cosh 2qt)((1 + 2q)cosh t + 2qt sinh t) + 2q(sinh 2qt)sinh(2q + 1)t/[q.sup.2].

Case v: For p = q = 0,

g'(t) = [partial derivative][g.sub.p, q](t)/[partial derivative]t = 4[t.sup.2]cosh t + 8t sinh t.

It is obvious that [partial derivative][g.sub.p, q](t)/[partial derivative]t is continuous with respect to p and q on R x R, again in view of

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

These arguments leads that the above five cases are continuous for all values of p and q. This completes the proof of lemma 3.2.

Lemma 3.3. [lim.sub.t [right arrow] 0, t > 0][t.sup.-3]g(t) = -4/3(p + q - 3). Proof. It is easy to check that first and second derivatives of g(t) = 0, at t = 0. In the case of pq(p - q) [not equal to] 0. Applying L-Hospital's rule (three times) yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Similarly, for p = 0, q [not equal to] 0,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];

for q = 0, p [not equal to] 0,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];

for p = q [not equal to] 0,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];

for p = q = 0,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

This completes the proof.

The proof of our main result stated below follows from the above lemmas:

Theorem 3.1. For fixed (p, q) [member of] R x R,

(1) Stolarsky's extended family type mean means [N.sub.r, s](a, b ; p, q) are Schur harmonic convex with respect to (a, b) if p + q - 3 [greater than or equal to] 0.

(2) Stolarsky's extended family type mean means [N.sub.r, s](a, b ; p, q) are Schur harmonic concave if p + q - 3 [less than or equal to] 0.

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K. M. Nagaraja ([dagger]) and Sudhir Kumar Sahu ([double dagger])

([dagger]) Department of Mathematics, JSS Academy of Technical Education, Uttarahalli-Kengeri Main Road, Bangalore-60, Karanataka, India

([double dagger]) Post Graduate, Department of Statistics, Sambalpur University, Sambalpur, Orissa-768019, India

E-mails: nagkmn@gmail.com drsudhir1972@gmail.com

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Author: | Nagaraja, K.M.; Sahu, Sudhir Kumar |
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Publication: | Scientia Magna |

Date: | Jun 1, 2013 |

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