Inclusion of the Generalized Bessel Functions in the Janowski Class.

1. Introduction

Let A denote the class of analytic functions f defined in the open unit disk D = {z : [absolute value of (z)] < 1} normalized by the conditions f(0) = 0 = f'(0) - 1. If f and g are analytic in D, then f is subordinate to g, written f(z) < g(z), if there is an analytic self-map w of D satisfying w(0) = 0 and f = g [omicron] u >. For -1 [less than or equal to] B < A [less than or equal to] 1, let P[A, B] be the class consisting of normalized analytic functions p(z) = 1 + [c.sub.1]z + ... in D satisfying

P(Z) < 1 + Az/1 + Bz. (1)

For instance, if 0 [less than or equal to] [beta] < 1, then P[1 - 2 [beta],-1] is the class of functions p(z) = 1 + [c.sub.1]z + ... satisfying Re p(z) > [beta] in D.

The class [S.sup.*][A, B] of Janowski starlike functions  consists of f [member of] A satisfying

Zf'(z)/f(z) [member of] P [A,B]. (2)

For 0 [less than or equal to] [beta] < 1, [S.sup.*][1-2[beta], -1] := [S.sup.*]([beta]) is the usual class of starlike functions of order [beta]; [S.sup.*][1 - [beta], 0] = [S.sup.*.sub.[beta]] = {f [member of] A : [absolute value of (zf'(z)/f(z) - 1)] < 1 - [beta]}; and [S.sup.*] [[beta], -[beta]] := [S.sup.*] [[beta]] = {f [member of] A : [absolute value of (zf'(z)/f(z)-1)] < [beta][absolute value of (z/'(z)/f(z)+1)]}. These classes have been studied, for example, in [2, 3]. A function f [member of] A is said to be close-to-convex of order [beta] [4,5] if Re (zf' (z)/g(z)) > [beta] for some g [member of] [S.sup.*] := [S.sup.*(0).]

This article studies the generalized Bessel function [u.sub.p](z) = [u.sub.p,b,c](z) given by the power series

[U.sub.p](z) = [sub.0][F.sub.1]([kappa], -c/4z) = [[infinity].summation over (k=0)] [(-1).sup.k][c.sup.k]/[4.sup.k][([kappa]).sub.k] [z.sup.k]/k!, (3)

where [kappa] = p + (b+1)/2 [not equal to] 0, -1, -2, -3, ... The function [u.sub.p](z) is analytic in D and solution of the differential equation

4[z.sup.2]u" (z) + 4[kappa]zu' (z) + czu (z) = 0, (4)

if b, p, c in C, such that [kappa] = p + (b + 1)/2 [not equal to] 0,-1,-2,-3, ..., and z [member of] D. This normalized and generalized Bessel function of the first kind of order p also satisfies the following recurrence relation:

4[kappa][u'.sub.p] (z) = -[cu.sub.p+1](z), (5)

which is useful tool to study several geometric properties of Up. There have been several works [6-11] studying geometric properties of the function [u.sub.p](z), for example, on its close-to convexity, star likeness, and convexity, and radius of star likeness and convexity.

In Section 2 of this paper, sufficient conditions on A, B, c, and [kappa] are determined that will ensure up satisfies the subordination [u.sub.p](z) < (1+Az)/(1+Bz). It is to be understood that a computationally intensive methodology with shrewd manipulations is required to obtain the results in this general framework. The benefits of such general results are that, by judicious choices of the parameters A and B, they give rise to several interesting applications, which include extending the results of previous works. Using this subordination result, sufficient conditions are obtained for (-4[kappa]/c)u (z) [member of] P[A, B], which next readily gives conditions for (-4[kappa]/c)([u.sub.p](z) - 1) to be close-to-convex. Section 3 gives emphasis to the investigation of [u.sub.p](z) to be Janowski convex as well as of z[u.sub.p](z) to be Janowski starlike. The following lemma is needed in sequel.

Lemma 1 (see [5,12]). Let [OMEGA] [subset] C and [PHI] : [C.sup.2] x D [right arrow] C satisfy

[PHI](i[rho],[alpha];z) [not member of] [OMEGA] (6)

whenever z [member of] D, [rho] is real, and [alpha] [less than or equal to] -(1 + [p.sup.2])/2.If p is analytic in D, with p(0) = 1, and [PHI](p(z), zp'(z); z) [member of] [OMEGA] for z [member of] D, then Re p(z) > 0 in D.

In the case [PHI] : [C.sup.3] x D [right arrow] C, then the condition in Lemma 1 is generalized to

[PHI](ip, [sigma], [mu] + iv; z) [not member of] [OMEGA]. (7)

[rho] is real, [sigma] + [mu] [less than or equal to] 0, and [sigma] [less than or equal to] -(1 + [[rho].sup.2])/2.

2. Close-To-Convexity of Generalized Bessel Functions

In this section, one main result on the close-to-convexity of the generalized Bessel function with several consequences is discussed in detail.

Theorem 2. Let -1 [less than or equal to] B [less than or equal to] 3 - 2[square root of 2] [approximately equal to] 0.171573. Suppose B < A [less than or equal to] 1 and c,[kappa] [member of] R satisfy

k - 1 [greater than or equal to] (1+B)(1+A)/4(A - B)[absolute value of (c)]. (8)

Further let A, B, k, and c satisfy either the inequality

[mathematical expression not reproducible] (9)

whenever

[absolute value of (2 (k - 1) (1 - B) (A + B) c + [(1 + B).sup.2] (1 + A) c)] [greater than or equal to] 1/2(a-b)(1-b)[c.sup.2] (10)

or the inequality

[mathematical expression not reproducible] (11)

whenever

[absolute value of (2 ([kappa] - 1) (1 - B) (A + B) c + [(1 + B).sup.2] (1 + A) c)] < 1/2(A - B)(1 - B)[c.sup.2]. (12)

If (1+B)[u.sub.p](z) [not equal to] (1 + A) for all z [member of] D, then up(z) [member of] P[A,B].

Proof. Define the analytic function p : D [right arrow] C by

P(Z) = (1-A)-(1-B) [u.sub.p](z)/(1 + A)-(1+B)[u.sub.p](z). (13)

Then, a computation yields

[mathematical expression not reproducible]. (14)

Thus, using identities (14), the Bessel differential equation (4) can be rewritten as

[mathematical expression not reproducible]. (15)

Assume [OMEGA] = {0}, and define [PHI](r, s, t; z) by [PHI] (r, s, t; z)

:= 2(1+B)/(1 - B) + (1 + B)r [S.sup.2] + [kappa]S ((1-B) + (1 + B)r)((1 - A) + (1 + A)r)/8(A - B) cz. (16)

It follows from (15) that [PHI](p(z),zp'(z),[z.sup.2]p"(z);z) [member of] [OMEGA]. To ensure Re p(z) > 0 for z [member of] D, from Lemma 1, it is enough to establish Re [PHI](i[rho], [sigma], [mu] + iv; z) < 0 in D for any real [rho], [sigma] [less than or equal to] -(1 + [[rho].sup.2])/2, and [sigma] + [mu] [less than or equal to] 0.

With z = x + iy [member of] D in (16), a computation yields

Re [PHI] (i[rho], [mu] + iv; z)

[mathematical expression not reproducible]. (17)

Since [sigma] [less than or equal to] -(1 + [[rho].sup.2])/2 and B [member of] [-1,3-2[square root of 2]],

[mathematical expression not reproducible]. (18)

Thus

Re [PHI](i[rho], [sigma], [mu] + iv; z)

[mathematical expression not reproducible], (19)

where

[mathematical expression not reproducible]. (20)

Condition (8) shows that

[mathematical expression not reproducible]. (21)

Since [max.sub.p[member of]R] {[p.sub.1][p.sup.2] + [q.sub.1][rho] + [r.sub.1]} = (4[p.sub.1][r.sub.1] - [q.sup.2.sub.1])/(4[p.sub.1]) for [p.sub.1] < 0, it is clear that Q([rho]) < 0 when

[mathematical expression not reproducible], (22)

with [absolute value of (y)][absolute value of (Y)] < 1. As [y.sup.2] < 1 - [x.sup.2], the above condition holds whenever

[mathematical expression not reproducible], (23)

that is, when

[mathematical expression not reproducible]. (24)

To establish inequality (24), consider the polynomial R given by

R (x) := m[x.sup.2] + nx + r, [absolute value of (x)] < 1, (25)

where

[mathematical expression not reproducible] (26)

Constraint (10) yields [absolute value of (n)] [greater than or equal to] 2[absolute value of (m)], and thus R(x) [greater than or equal to] m + r-[absolute value of (n)]. Now inequality (9) readily implies that

[mathematical expression not reproducible]. (27)

Now consider the case of constraint (12), which is equivalent to [absolute value of (n)] < 2m. Then the minimum of R occurs at x = -n/(2m), and (11) yields

R(x) [greater than or equal to] 4mr - [n.sup.2]/4m [greater than or equal to] 0. (28)

Evidently [PHI] satisfies the hypothesis of Lemma 1, and thus Re p(z) > 0; that is,

(1-A)-(1-B)[u.sub.p] (z)/(1+A)-(1 + B)up (z) < 1-z/1 -z. (29)

Hence there exists an analytic self-map w of D with w(0) = 0 such that

(1-A)-(1-B)[u.sub.p] (z)/(1 + A)-(1 + B)[u.sub.p] (z) = 1+w(z)/1-w(z), (30)

which implies that [u.sub.p](z) < (1 + Az)/(1 + Bz).

Theorem 2 gives rise to simple conditions on c and [kappa] to ensure [u.sub.p](z) maps D into a half plane.

Corollary 3. Let c [less than or equal to] 0 and [kappa] [greater than or equal to] max{5/4,1 + [c.sup.2]/2}. Then Re [u.sub.p](z) > c/(c - 1).

Proof. Choose A = -(c + 1)/(c - 1) and B = - 1 in Theorem 2. Then condition (8) is equivalent to [kappa] [greater than or equal to] 1 and (10) reduces to [kappa] [greater than or equal to] 5/4, and clearly both hold for [kappa] [greater than or equal to] max{5/4,1 + [c.sup.2]/2}. The proof will complete if hypothesis (9) holds; that is,

[([kappa]-1).sup.2] [greater than or equal to] 1/2([kappa]-1)[c.sup.2]. (31)

Since [kappa] [greater than or equal to] 1 + [c.sup.2]/2, it follows that

[([kappa]-1).sup.2] - 1/2([kappa] - 1)[c.sup.2] = ([kappa] - 1) ([kappa] - 1 - [c.sup.2]/2) [greater than or equal to] 0, (32)

which establishes (31).

Corollary 4. Let c [member of] R. Then if [kappa] [greater than or equal to] [absolute value of (c)]/2, then Re [u.sub.p](z) > 1/2.

Proof. Put A = 0 and B = -1 in Theorem 2. Condition (8) reduces to [kappa] [greater than or equal to] 1, which holds in all cases. It is sufficient to establish conditions (10) and (9) or, equivalently,

4([kappa] - 1) - [absolute value of (c)] [greater than or equal to] 0, [([kappa] -1).sup.2] - 1/2 ([kappa]-1)[absolute value of (c)] [greater than or equal to] 0. (33)

The hypothesis [kappa] - 1 [greater than or equal to] [absolute value of (c)]/2 implies that 4([kappa]-1)-[absolute value of (c)] [greater than or equal to] [absolute value of (c)] [greater than or equal to] 0 and [([kappa] -1).sup.2] - (1/2)([kappa] - 1)[absolute value of (c)] [greater than or equal to] ([kappa] - 1)([kappa] - 1 - [absolute value of (c)]/2) [less than or equal to] 0.

Next theorem gives the sufficient condition for close-toconvexity when B [greater than or equal to] 3 - [square root of 2]

Theorem 5. Let 3-2[square root of 2] [less than or equal to] B < A [less than or equal to] 1 and c,k [member of] R satisfys

[kappa] - 1 [greater than or equal to] (1 + B)(1 + A)/4(A - B) [absolute value of (C)]. (34)

Suppose A, B, [kappa], and c satisfy either the inequality

[mathematical expression not reproducible] (35)

whenever

[mathematical expression not reproducible] (36)

or the inequality

[mathematical expression not reproducible] (37)

whenever

[absolute value of (([kappa] -1) (A + B) [(1 + B).sup.3] + 8B(1 - [B.sup.2]) (1 + A)) c)] < [c.sup.2]/4 (A - B) [(1 + B).sup.3]. (38)

If (1 + B)[u.sub.p](z) [not equal to] (1 + A) for all z [member of] D, then [u.sub.p](z) [member of] P[A,B].

Proof. First, proceed similarly to the proof of Theorem 2 and derive the expression of Re [PHI](i[rho], [sigma], [mu] + iv;z) as given in (17). Now, for [sigma] [less than or equal to] - (1 + [[rho].sup.2])/2, [rho] [member of] R, and B [greater than or equal to] 3-2 [square root of 2],

[mathematical expression not reproducible], (39)

and then, with z = x + iy [member of] D and [mu] + [sigma] < 0, it follows that

[mathematical expression not reproducible], (40)

where

[mathematical expression not reproducible], (41)

Observe that inequality (34) implies that [p.sub.2] < 0. Thus [Q.sub.1]([rho]) < 0 for all [rho] [member of] R provided [q.sup.2.sub.2] < 4[p.sub.2][r.sub.2]; that is, for [absolute value of (x)], [absolute value of (y)] < 1,

[mathematical expression not reproducible]. (42)

With [y.sup.2] < 1 - [x.sup.2], it is enough to show, for [absolute value of x] < 1,

[mathematical expression not reproducible], (43)

which is equivalent to

[R.sub.1](x) := [m.sub.1][x.sup.2] + [n.sub.1]x + [r.sub.1] [greater than or equal to] 0, (44)

where

[mathematical expression not reproducible]. (45)

If (36) holds, then [absolute value of ([n.sub.1])] [greater than or equal to] 2[absolute value of ([m.sub.1])] and [R.sub.1](x) [greater than or equal to] [m.sub.1] + [r.sub.1] - [absolute value of ([n.sub.1])], which is nonnegative from (35). On the other hand, if (38) holds, then [absolute value of ([n.sub.1])] < 2[absolute value of ([m.sub.1])], [R.sub.1](x) [greater than or equal to](4[m.sub.1][r.sub.1] - [n.sup.2.sub.1])/4[m.sub.1], and (37) implies [R.sub.1](x) [greater than or equal to] 0. Either case establishes (44).

Theorem 6. Let -1 [less than or equal to] B [less than or equal to] 3 - 2 [square root of 2] [approximately equal to] 0.171573. Suppose B < A [less than or equal to] 1 and c,k [member of] R with c [not equal to] 0 satisfying

[kappa] [greater than or equal to] (1 + B)(1+A)/4 (A - B)[absolute value of (c)]. (46)

Further let A, B, k, and c satisfy either

[mathematical expression not reproducible] (47)

whenever

[absolute value of (2[kappa] (1 - B) (A + B) c + [(1+ B).sup.2] (1 + A) c)] [greater than or equal to] - 1/2 (A - B)(1 - B)[c.sup.2] (48)

or the inequality

[mathematical expression not reproducible] (49)

when

[absolute value of (2[kappa] (1 - B) (A + B) c + [(1+ B).sup.2] (1 + A) c)] < 1/2(A - B)(1 - B)[c.sup.2]. (50)

If (1 + B)[U.sub.p](z)[not equal to] (1 + A) for all z [member of] D, then (-4k/c)[u'.sub.p] (z) [member of] P[A, B].

Theorem 7. Let 3 - 2[square root of 2 < B < A [less than or equal to] 1. Suppose c,[kappa] [member of] R and a [not equal to] 0, such that

[kappa] [greater than or equal to] (1 + B)(1 + A)/4(A - B) [absolute value of (c)]. (51)

Suppose A, B, [kappa], and c satisfy either

[mathematical expression not reproducible] (52)

whenever

[absolute value of (k [(1 + B).sup.3] (A + B) c + 8B (1 - [B.sup.2]) (1 + A) c)] [greater than or equal to] [c.sup.2]/4 (A - B) [(1 + B).sup.3] (53)

or the inequality

[mathematical expression not reproducible] (54)

when

[absolute value of (2[kappa][(1 + B).sup.3] (A + B) c + 8B (1 - [B.sup.2]) (1 + A) c)] < [c.sup.2]/4 (A - B)[(1 + B).sup.3].

If (1 + B)[U.sub.p](z) [not equal to] (1 + A) for all z [member of] D, then (-4[kappa]/c)[u'.sub.p](z) [member of] P[A,B].

Corollary 8. Let c [less than or equal to] -1 and

k [greater than or equal to] max {c(c + 1)/2}, c/(c + 1). (56)

Then (-4[kappa]/c)([u.sub.p](z) - 1) is close-to-convex of order (c + 1)/c with respect to the identity function.

Corollary 9. Let c be a nonzero real number and k [greater than or equal to] [absolute value of (c)]/2. Then

Re (-4[kappa]/c)[u'.sub.p] (z) > 1/2. (57)

3. Janowski Starlikeness of Generalized Bessel Functions

This section contributes to finding conditions to ensure a normalized and generalized Bessel function z[u.sub.p](z) in the class of Janowski starlike functions. For this purpose, first sufficient conditions for [u.sub.p]s (z) to be Janowski convex are determined, and then an application of relation (5) yields conditions for z[u.sub.p](z) [member of] [S.sup.*][A, B].

Theorem 10. Let c,[kappa] [member of] R be such that (A - B)[u'.sub.p](z) [not equal to] (1 + B) z[U".sub.p](z) for all z [member of] D and -1 [less than or equal to] B < A [less than or equal to] 1. Supposes

[kappa](1 + b) [greater than or equal to] [(1 + B).sup.2]/4(A - B))[absolute value of (c)] -(1 + A - B). (58)

Further let A, B, [kappa], and c satisfy

[mathematical expression not reproducible]. (59)

If 0 [not member of] [u'.sub.p](D) and 0 [not member of] [u".sub.p](D), then

1 + Z[u.sub."p](z)/[u'.sub.p](z) < 1 + Az/1 + Bz. (60)

Proof. Define an analytic function p : D [right arrow] C by

p(z) := (A-B)[u'.sub.p] (z) + (1 - B)z[u".sub.p](z)/(A-B)[u'.sub.p](z)-(1 + B)z[u".sub.p](z). (61)

Then

[mathematical expression not reproducible] (62)

[mathematical expression not reproducible]. (63)

A rearrangement of (63) yields

z[u".sub.p](z)/[U".sub.p](z) = 2zp'(z)(p(z]-1)((p(z) + 1) + B(p(z)-1)) - 1 + z[u".sub.p](z)/u'p(z). (64)

Thus,

[mathematical expression not reproducible]. (65)

Now a differentiation of (4) leads to

4[z.sup.2][u'".sub.p](z) + 4 ([kappa] + 1) z[u".sub.p] (z) + cz[u'.sub.p] (z) = 0, (66)

which gives

[mathematical expression not reproducible] (67)

Using (62) and (65), (67) yields

[mathematical expression not reproducible], (68)

and equivalently

[mathematical expression not reproducible]. (69)

Define

[PHI][(p(z), zp' (z), z) := zp' (z) + [F.sub.1] (p (z)).sup.2] + [F.sub.2]p(z) + [F.sub.3],

where

[mathematical expression not reproducible], (71)

Thus, (69) yields [PHI](p(z),zp'(z),z) [member of] [OMEGA] = {0}. Now, with z = x + iy [member of] D,let

[mathematical expression not reproducible]. (72)

[mathematical expression not reproducible], (73)

Note that condition (58) implies (1 + 2[G.sub.1])/2 > 0. In this case, Q has a maximum at [rho] = [G.sub.2]/(1 + 2[G.sub.1]). Thus Q([rho]) < 0 for all real p provided

[G.sup.2.sub.2] [less than or equal to] (1 + 2[G.sub.1])(1-2[G.sub.3]), [absolute value of (x)], [absolute value of (y)] < 1. (74)

Since [y.sup.2] < 1 - [x.sup.2], it is left to show that

[mathematical expression not reproducible], (75)

[absolute value of (x)] < 1. The above inequality is equivalent to

H (x) = [h.sub.2] (A, B)x + [h.sub.3] (A, B) > 0, (76)

International Journal of Analysis

where

[mathematical expression not reproducible], (77)

Since [absolute value of (x)] < 1, the left-hand side of inequality (76) satisfies

[h.sub.2] (A,B)x + [h.sub.3] (A,B) [greater than or equal to] - [absolute value of ([h.sub.2])] (A,B) + [h.sub.3] (A,B). (78)

Now it is evident from (59) that H(x) [greater than or equal to] 0 which establishes inequality (76).

Thus [phi] satisfies the hypothesis of Lemma 1, and hence Re p(z) > 0, or equivalently

(A-B)[u'.sub.p] + (1-B)z[u".sub.p]/(A-B) [u'.sub.p] -(1 + B)z[u".sub.p] < 1+z/1-z (79)

By definition of subordination, there exists an analytic selfmap w of D with w(0) = 0 and

(A-B)[u'.sub.p] (z) + (1- B) z[u'.sub.p] (z)/(A-B) [u'.sub.p] (z)-(1 + B) z[u".sub.p] 1+w(z)/(z) 1-w(z). (80)

A simple computation shows that

1 + z[u".sub.p] (z)/[u'.sub.p](z) = 1 + Aw(z)/1 + Bw(z), (81)

and hence

1 + Z[u".sub.p](z)/[u'.sub.p](z) < 1 + Az/1 + Bz. (82)

Relation (5) also shows that

z(z[u.sub.P] (z))'x/Z[U.sub.p](z) = 1 + [u".sub.p-1](z)/ [u'.sub.p-1](z). (83)

Together with Theorem 10, relation (83) immediately yields the following result for z[u.sub.p](z) [member of] [S.sup.*] [A, B].

Theorem 11. Let c and k be real numbers such that (A - B)[u'.sub.p-1] (z) = (1+B) z[U'.sub.p-1](z) for all z [member of] D and -1 < B < A [less than or equal to] 1.

Suppose

[kappa](1 + b)[greater than or equal to] [(1 + B).sup.2]/4(A - B) [absolute value of (c)] -(a-2B). (84)

Further let A, B, k, and c satisfy

[mathematical expression not reproducible]. (85)

Then z[u.sub.p](z) [member of] [S.sup.*][A,B].

http://dx.doi.org/10.1155/2016/4740819

Competing Interests

The authors declare that they have no competing interests.

References

 W. Janowski, "Some extremal problems for certain families of analytic functions. I," Annales Polonici Mathematici, vol. 28, pp. 297-326, 1973.

 R. M. Ali, V. Ravichandran, and N. Seenivasagan, "Sufficient conditions for Janowski starlikeness," International Journal of Mathematics and Mathematical Sciences, vol. 2007, Article ID 62925, 7 pages, 2007.

 R. M. Ali, R. Chandrashekar, and V. Ravichandran, "Janowski starlikeness for a class of analytic functions," Applied Mathematics Letters, vol. 24, no. 4, pp. 501-505, 2011.

 A. W. Goodman, Univalent Functions, vol. 1-2, Mariner, Tampa, Fla, USA, 1983.

 S. S. Miller and P. T. Mocanu, Differential Subordinations, Monographs and Textbooks in Pure and Applied Mathematics, vol. 225, Dekker, New York, NY, USA, 2000.

 A. Baricz, "Geometric properties of generalized Bessel functions," Publicationes Mathematicae Debrecen, vol. 73, no. 1-2, pp. 155-178, 2008.

 A. Baricz, "Geometric properties of generalized Bessel functions of complex order," Mathematica, vol. 48, no. 71, pp. 13-18, 2006.

 A. Baricz and S. Ponnusamy, "Starlikeness and convexity of generalized Bessel functions," Integral Transforms and Special Functions, vol. 21, no. 9-10, pp. 641-653, 2010.

 A. Baricz and R. Szasz, "The radius of convexity of normalized Bessel functions of the first kind," Analysis and Applications, vol. 12, no. 5, pp. 485-509, 2014.

 V. Selinger, "Geometric properties of normalized Bessel functions," Pure Mathematics and Applications, vol. 6, no. 2-3, pp. 273-277, 1995.

 R. Szasz and P. A. Kupan, "About the univalence of the Bessel functions," Studia Universitatis Babes-Bolyai Mathematica, vol. 54, no. 1, pp. 127-132, 2009.

 S. S. Miller and P. T. Mocanu, "Differential subordinations and inequalities in the complex plane," Journal of Differential Equations, vol. 67, no. 2, pp. 199-211, 1987.

Saiful R. Mondal and Mohammed Al Dhuain

Department of Mathematics and Statistics, Collage of Science, King Faisal University, Hofuf, Al-Hasa 31982, Saudi Arabia

Correspondence should be addressed to Saiful R. Mondal; smondal@kfu.edu.sa

Received 19 April 2016; Accepted 4 October 2016

Title Annotation: Printer friendly Cite/link Email Feedback Research Article Mondal, Saiful R.; Dhuain, Mohammed Al International Journal of Analysis Report Jan 1, 2016 3971 Straightforward Proofs of Ostrowski Inequality and Some Related Results. Variational-Like Inequalities for Weakly Relaxed [eta]-[alpha] Pseudomonotone Set-Valued Mappings in Banach Space. Bessel functions Mathematical research