# An integral operator and its application on certain classes of analytic functions.

1. Introduction

Let A be the class of functions

f(z) = z + [[infinity].summation over (n=2)] [a.sub.n][z.sup.n], (1.1)

which are analytic in the unit discE = {z : [absolute value of z] < 1}. We denote [S.sup.*]([alpha]), C([alpha]) and K([beta],[alpha]) be the subclasses of A consisting of functions which are respectively, starlike of order [alpha], convex of order [alpha] and close-to-convex of order [beta] and type [alpha] in E.

We consider the following integral operator

[L.sup.[mu].sub.[lambda]]: A [right arrow] A, for [lambda] > -1; [mu] > 0; f [member of] A, defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.2)

where [GAMMA] denotes the gamma function and f(z) is given by (1. 1). From (1.2), we can obtain the generalized Bernadi operator as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

From (1.2), it can be seen that

z([L.sup.[mu]+1.sub.[lambda]]f(z))' = ([lambda] + [mu] + 1)[L.sup.[mu].sub.[lambda]]f(z) - ([lambda] + [mu])[L.sup.[mu]+1.sub.[lambda]]f(z). (1.3)

The class A is closed under the Hadamard product or convolution, defined by

to

([f.sub.1] * [f.sub.2])(z) = z + [[infinity].summation over (n=2)][a.sub.n][b.sub.n][z.sup.n],

where

[f.sub.1](z) = z + [[infinity].summation over (n=2)][a.sub.n][z.sup.n] and [f.sub.2](z) = z + [[infinity].summation over (n=2)][b.sub.n][z.sup.n].

Let [P.sub.k]([alpha]) be the class of functions p(z) analytic in the unit disc E, satisfying the properties p(0) = 1 and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.4)

where z = [re.sup.i[theta]], k [greater than or equal to] 2 and 0 [less than or equal to] [alpha] < 1. For [alpha] = 0, we obtain the class [P.sub.k] defined by Pinchuk in .

We can also represent p [member of] [P.sub.k]([alpha]) as

P(z) = (k/4 + 1/2)[p.sub.1](z) - (k/4 - 1/2)[p.sub.2](z), (1.5)

where [p.sub.i] [member of] P([alpha]), for i = 1,2 and z [member of] E. We have the following known classes.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Remark 1.1

f [member of] [V.sub.k]([alpha]) [??] zf' [member of] [R.sub.k]([alpha]).

Using the integral operator [L.sup.[mu].sub.[lambda]], we introduce the following classes.

Definition 1.1. [R.sup.k]([lambda], [mu], [alpha]) = {f : f [member of] A and [L.sup.[mu].sub.[lambda]] f(z) [member of] [R.sup.k]([alpha]), 0 [less than or equal to] [alpha] < 1}.

Definition 1.2. [V.sub.k] ([lambda], [mu], [alpha]) = {f : f [member of] A and [L.sup.[mu].sub.[lambda]] f(z) [member of] [V.sub.k]([alpha]), 0 [less than or equal to] [alpha] < 1}.

Definition 1.3. [T.sup.*.sub.k] ([lambda], [mu], [beta], [alpha]) = {f : f [member of] A and [L.sup.[mu].sub.[lambda]] f(z) [member of] [T.sup.*.sub.k]([beta], [alpha]), 0 [less than or equal to] [alpha], 0 < 1}.

Definition 1.4. [P'.sub.k]([lambda], [mu], [alpha]) = {f : f [member of] A and [L.sup.[mu].sub.[lambda]] f(z) [member of] [P'.sub.k]([alpha]), 0 [less than or equal to] [alpha] < 1}.

2. Preliminary results

We shall need the following result due to Miller and Mocanu .

Lemma 2.1. Let u = [u.sub.1] + [iu.sub.2] and v = [v.sub.1] + [iv.sub.2] and let [PHI] be a complex-valued function satisfying the conditions:

(i) [PHI](u,v) is continuous in D [subset] [C.sup.2],

(ii) (1,0) [member of] D and Re[PHI](1,0) > 0,

(iii) Re[PHI]([iu.sub.2], [v.sub.1]) [less than or equal to] 0, whenever ([iu.sub.2], [v.sub.1]) [member of] D and [v.sub.1] [less than or equal to] - 1/2(1 + [u.sup.2.sub.2]).

If h(z) = 1 + [[infinity].summation over (m=2)][c.sub.m][z.sup.m] is a function analytic in E such that (h(z),zh'(z)) [member of] D and Re[PHI](h(z), zh'(z)) > 0, for z [member of] E, then Reh(z) > 0 in E.

Lemma 2.2. Let p(z) be analytic in E with p(0) = 1 and Re p(z) > 0, z [member of] E. Then for s > 0 and [eta] [not equal to] -1 (complex),

Re{p(z) + szp'(z)/p(z) + [eta]} > 0, for [absolute value of z] < [r.sub.0],

where [r.sub.0] is given by

[r.sub.0] = [absolute value of [eta] + 1]/[square root of A + [([A.sup.2] - [absolute value of [[eta].sup.2] - 1]).sup.1/2]], A = 2[(s + 1).sup.2] + [[absolute value of [eta]].sup.2] - 1

and this result is best possible. For this result we refer to .

Lemma 2.3.. Let [PSI] be convex and g be starlike in E. Then, for F analytic in E with F(0) = 1, [PSI] * Fg/[PSI] * g is contained in the convex hull of F (E).

Lemma 2.4.. If p(z) is analytic in E with p(0) = 1, and if [lambda] is a complex number satisfying Re[lambda] [greater than or equal to] 0, ([lambda] [not equal to] 0), then Re[p(z) + [lambda]zp'(z)] > [beta], (0 [less than or equal to] [beta] < 1) implies Re p(z) > [beta] + (1 - [beta])(2[gamma] -1), where [gamma] is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

which is an increasing function of Re [lambda] and 1/2 [less than or equal to] [gamma] [less than or equal to] 1. The estimate is sharp in the sense that bound cannot be improved.

3. Main Results

Theorem 3.1. Let [lambda] > -1, [mu] > 0 and [alpha] [member of] [[[alpha].sub.0],1] with

[[alpha].sub.0] = Max{-[lambda] + [mu]/2, - ([lambda] + [mu]}. Then [R.sup.k]([lambda], [mu], [alpha]) [subset] [R.sup.k]([lambda], [mu] + 1, [beta]),

where

[beta] = 1 + ([lambda] + [mu])/[sub.2][F.sub.1](2(1 - [alpha]),1,2 + [lambda] + [mu];1/2) - ([lambhda] + [mu]), (3.1)

and [sub.2][F.sub.1] is hypergeometric function and this result is sharp.

Proof. Let f [member of] [R.sub.k]([lambda], [mu], [alpha]) and let

z([L.sup.[mu]+1.sub.[lambda]]f(z))'/[L.sup.[mu]+1.sub.[lambda]]f(z) = H(z), (3.2)

where h(z) = 1 + [c.sub.1]z + [c.sub.2][z.sup.2] + [c.sub.3][z.sup.3] + ...is analytic and h(0) = 1 in E.

From (1.3) and (3.2), we have

([lambda] + [mu] +1) [L.sup.[mu].sub.[lambda]]f(z)/[L.sup.[mu]+1.sub.[lambda]] = H(z) + ([lambda] + [mu]).

By logarithmic differentiation, we have

z([L.sup.[mu].sub.[lambda]]f(z))'/[L.sup.[mu].sub.[lambda]]f(z) = H(z) + [zH'(z)H(z) + [lambda] + [mu]][member of] [P.sub.k]([alpha]), z [member of] E.

Let

[[PSI].sub.[lambda],[mu]](z) = ([lambda] + [mu]/[lambda] + [mu] + 1)z/1 - z + (1/[lambda] + [mu] + 1)z/[(1 - z).sup.2]

and

H(z) = (k/4 + 1/2)[h.sub.1](z) - (k/4 - 1/2)[h.sup.2](z).

We want to show that that H [member of] [P.sub.k]([beta]), where [beta] is given by (3.1) or equivalently [h.sub.i] [member of] P([beta]), i = 1,2.

Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where {[h.sub.i](z) + [zh.sub.i]'(z)/[h.sub.i](z) + [lambda] + [mu]}[member of] P([alpha]), i = 1,2.

We now use a result [3, Theorem 3.3e] to obtain that [h.sub.i] [member of] P([beta]), where [beta] as given by (3.1). Hence H(z) [member of] [P.sub.k]([beta]) and consequently f [member of] [R.sub.k]([lambda], [mu] + 1, [beta]).

Sharpness is given by the function

g(z) = [alpha]/g(z) - ([lambda] + [mu])

and g(z)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Special cases.

(i) For [alpha] = 0, we obtain a result given in .

(ii) For k = 2 and [beta] = [alpha], we have

[R.sub.2]([lambda], [mu], [alpha]) [subset] [R.sup.2]([lambda], [mu] + 1, [alpha]), that is [S.sup.*.sub.[lambda],[mu]]([alpha]) [subset] [S.sup.*.sub.[lambda],[mu]+1]([alpha]), see .

Theorem 3.2. For [lambda] > -1, [mu] > 0. Then [V.sub.k]([lambda], [mu], [alpha]) [subset] [V.sub.k]([lambda], [mu] +1,[beta]), where [beta] is given by (3.1).

Proof. Let f [member of] [V.sub.k]([lambda], [mu], [alpha]). Then [L.sup.[mu].sub.[lambda]] f(z) [member of] [V.sub.k]([alpha]) and, by using Remark 1.1, we have z([L.sup.[mu].sub.[lambda]]f(z))' [member of] [R.sub.k]([alpha]).

This implies

[L.sup.[mu].sub.[lambda]](zf'(z)) [member of] [R.sub.k]([alpha]) ===> zf'(z) [member of] [R.sub.k]([lambda], [mu], [alpha]) [subset] [R.sub.k]([lambda], [mu] + 1, [beta]).

Consequently f [member of] [V.sub.k]([lambda], [mu] + 1, [beta]), where [beta] is given by (3.1).

Theorem 3.3. Let [lambda] > -1, [mu] > 0 and [lambda] + [mu] > -[alpha]. Then [T.sup.*.sub.k] ([lambda], [mu], [beta], [alpha]) [subset] [T.sup.*.sub.k]([lambda], [mu] + 1, [beta], [alpha]), where 0 [less than or equal to] [alpha], [beta] < 1.

Proof. Let f [member of] [T.sup.*.sub.k] ([lambda], [mu], [beta], [alpha]). Then there exists [g.sub.1](z) [member of] [R.sub.2]([alpha]) such that

z([L.sup.[mu].sub.[lambda]]f(z))/[g.sub.1](z) [member of] [P.sub.k]([beta]). (3.3)

Let [g.sub.1](z) = [L.sup.[mu].sub.[lambda]]g(z). Then g(z) [member of] [R.sub.2]([lambda], [mu], [alpha]).

We set

z([L.sup.[mu].sub.[lambda]]f(z))'/[L.sup.[mu]+1.sub.[lambda]] = H(z) = (1 - [beta])h(z) + [beta], (3.4)

where h(z) = 1 + [c.sub.1]z + [c.sub.2][z.sup.2] + [c.sub.3][z.sup.3] +... is analytic and h(0) = 1 in E.

By using (1.3) and after some simplification, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.5)

Also, g [member of] [R.sub.2]([lambda], [mu], [alpha]) and by using Theorem 3.1, with k = 2 and [beta] = [alpha], we have g [member of] [R.sub.2]([lambda],[mu] + 1, [alpha]). Therefore, we can write

z([L.sup.[mu]+1.sub.[lambda]]g(z))'/[L.sup.[mu]+1.sub.[lambda]]g(z) = [H.sub.0](z) = (1 - [alpha])q(z) + [alpha], where q(z) [member of] P.

By logarithmic differentiation of (3.4) and after some simplification, we have

z([L.sup.[mu]+1.sub.[lambda]](zf'(z)))'/[L.sup.[mu].sub.[lambda]]g(z) = H(z)[H.sub.0](z) + zH'(z) (3.6)

From (3.5) and (3.6), we obtain

z([L.sup.[mu].sub.[lambda]]f(z))'/[L.sup.[mu].sub.[lambda]]g(z) = H(z) + zH'(z)/[H.sub.0](z) + [lambda] + [mu] [member of] [P.sub.k]([beta]). (3.7)

Let

H(z) = (k/4 + 1/2) {(1 - [beta])[h.sub.1](z) + [beta]}-(k/4 - 1/2) {(1 - [beta])[h.sub.2](z) + [beta]}

and

c(z) = [H.sub.0](z) + [lambda] + [mu] = (1 - [alpha]))(z) + [alpha] + [lambda] + [mu].

We want to show that H [member of] [P.sub.k]([beta]) or equivalently [h.sub.i] [member of] P for i = 1,2. Then Re c(z) > 0 if [lambda] + [mu] > -[alpha].

From (3.4) and (3.7), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and this implies that

Re{(1 - [beta])[h.sub.i](z) + (1 - [beta])[zh'.sub.i](z)/(1 - [alpha])q(z) + [alpha] + [lambda] + [mu]} > 0, z [member of] E, i = 1,2.

We formulate a functional [PHI](u,v) by taking u = [h.sub.i](z) and v = [zh'.sub.i](z). Thus

[PHI](u,v) = (1 - [beta])u + (1 - [beta])v/(1 - [alpha])q(z) + [alpha] + [lambda] + [mu].

It can easily seen that [PHI](u,v) satisfies the conditions (i) and (ii) of Lemma 2.1. To verify the condition(iii), we proceed, with q(z) = [q.sub.1] + [iq.sub.2], as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

When we put [v.sub.1] [less than or equal to] -1/2(1 + [u.sup.2.sub.2]), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

By applying lemma 2.1, we have Re [h.sub.i](z) > 0, for i = 1,2, and consequently h(z) [member of] [P.sub.k]. Thus f [member of] [T.sup.*.sub.k] ([lambda], [mu] + 1, [beta], [alpha]).

Theorem 3.4. Let [phi] be a convex function and f [member of] [R.sub.2]([lambda], [mu], [alpha]). Then [phi] * f [member of] [R.sub.2]([lambda], [mu], [alpha]).

Proof. Let G = [phi] * f and let

[phi](z) = z + [[infinity].summation over (n=2)][b.sub.n][z.sup.n] and f(z) = z + [[infinity].summation over (n=2)][a.sub.n][z.sup.n].

Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (3.8)

Also, f [member of] [R.sub.2]([lambda], [mu], [alpha]). Therefore, [L.sup.[mu].sub.[lambda]]f(z) [member of] [R.sub.2]([alpha]) = [S.sup.*]([alpha]).

By logarithmic differentiation of (3.8), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where F(z) = z([L.sup.[mu].sub.[lambda]]f(z))'/[L.sup.[mu].sub.[lambda]]f(z) is analytic in E and F(0) = 1. From Lemma 2.3, we see that z([L.sup.[mu].sub.[lambda]]G(z))'/[L.sup.[mu].sub.[lambda]]G(z) is contained in the convex hull of F(E). Since z([L.sup.[mu].sub.[lambda]]G(z))'/[L.sup.[mu].sub.[lambda]]G(z) is analytic in E and

F(E) [subset] [OMEGA] = {W: [z([L.sup.[mu].sub.[lambda]]W(z))'/[L.sup.[mu].sub.[lambda]]W(z)][member of] [p.sub.2](alpha)}

then z([L.sup.[mu].sub.[lambda]]G(z))'/[L.sup.[mu].sub.[lambda]]G(z) lies in [OMEGA]. This implies that G = [phi] * f [member of] [R.sub.2]([lambda], [mu], [alpha]).

Applications of Theorem 3.4

The class [R.sub.2]([lambda], [mu], [alpha]) is invariant under the following integral operators

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The proof immediately follows from Theorem 3.4. Since we can write [f.sub.i] = f * [[phi].sub.i], for with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for i = 1,2,3,4.

Theorem 3.5. Let for z [member of] E, f [member of] [R.sub.k]([lambda], [mu] + 1, [beta]). Then f [member of] [R.sub.k]([lambda], [mu], [beta]), for

[absolute value of z]< [r.sub.0] = [absolute value of [eta] + 1]/[square root of A + [([A.sup.2] - [absolute value of [[eta].sup.2] - 1]).sup.1/2]] (3.9)

where

A = 2[(s + 1).sup.2] + [absolute value of [[eta].sup.2]] - 1, [eta] = [lambda] + [mu], and s = 1. The value of [r.sub.0] is exact.

Proof. Let

z([L.sup.[mu]+1.sub.[lambda]]f(z))'/[L.sup.[mu]+1.sub.[lambda]]f(z) = h(z) = (k/4 + 1/2)[h.sub.1](z) - (k/4 - 1/2)[h.sub.2](z),

where h(z) = 1 + [c.sub.1]z + [c.sub.2][z.sup.2] + [c.sub.3][z.sup.3] + ...is analytic with h(0) = 1.

Using (1.3) and working in the same way as in Theorem 3.1, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where Re [h.sub.i](z) > 0, for i = 1,2. Therefore, by using Lemma 2.2. with [eta] = [lambda] + [mu] and s = 1, we have

Re{[h.sub.i](z) + [szh'.sub.i](z)/[h.sub.i](z) + [eta]} > 0, for [absolute value of z] < [r.sub.0],

and [r.sub.0] is given by (3.9). Hence z([L.sup.[mu].sub.[lambda]]f(z))'/[L.sup.[mu].sub.[lambda]]f(z) [member of] [P.sub.k] and consequently f [member of] [R.sub.k]([lambda], [mu], [beta])for [absolute value of z] < [r.sub.0].

Special case

For k = 2, [lambda] = 1, [mu] = 1, we have

f(z) [member of] [R.sub.2](1,2,0) [??] f(z) [member of] [R.sub.2](1,1,0) for [absolute value of z] < 0.8514,

that is

f(z) [member of] [R.sub.2](1,2,0) [??] [J.sub.1]f(z) [member of] [S.sup.*] for [absolute value of z] < 0.8514.

Theorem 3.6. Let [lambda] > -1, [mu] > 0. Then

[P'.sub.k]([lambda], [mu], [alpha]) [subset] [P'.sub.k]([lambda], [mu] + 1, [beta]),

where

[beta] = [alpha] + (1 - [alpha])(2[gamma] - 1), (3.10)

and [gamma] is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

which is an increasing function of 1/[lambda] + [mu] + 1 and 1/2 [less than or equal to] [gamma] [less than or equal to] 1.

Proof. We set

([L.sup.[mu]+1.sub.[lambda]]f(z))' = H(z) = (k/4 + 1/2)[h.sub.1](z) - (k/4 - 1/2)[h.sub.2](z).

where h(z) = 1 + [c.sub.1]z + [c.sub.2][z.sup.2] + [c.sub.3][z.sup.3] + ...is analytic and h(0) = 1. We want to show that H [member of] [P.sub.k]([beta]) or equivalently [h.sub.i] [member of] P([beta]) for i = 1,2.

By differentiating (1.3) and after some simplification, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where

[[lambda].sub.1] = 1/[lamnda] + [mu] + 1

Since ([L.sup.[mu].sub.[lambda]]f(z)) [member of] [P.sub.k]([alpha]), this implies Re([h.sub.i](z) + [[lambda].sub.1][zh'.sub.i](z)) > [alpha] and by using Lemma 2.4, we have Re [h.sub.i](z) > [beta], where [beta] is given by (3.10.). This implies that H(z) [member of] [P.sub.k]([beta]) and hence f [member of] [P'.sub.k] ([lambda], [mu] + 1, [beta]).

References

 C. Y. Gao, S. Yuan and H. M. Srivastava, Some functional inequalities and inclusion relationship associated with certain families of integral operator, Comput. Math. App. 49(2005), 1787-1795.

 S.S. Miller, Differential inequalities and caratheodory functions, Bull. Amer. Math. Soc. 81(1975), 79-81.

 S.S. Miller and P.T. Mocanu, Differential Subordination Theory and Applications, Pure and Applied Mathematics, Vol. 225, Marcel Dekker, New York, 2000.

 K.I. Noor, On analytic functions related to certain family of integral operators, J. Inequal. Pure and App. Math., 7(2) Art. 69 (2006).

 B. Pinchuk, Function with bounded boundary rotation, Isr. J. Math., 10(1971), 7-16.

 S. Ruscheweyh and T. Shiel-small, Hadamard product of schlicht functions and polya-schoenberg conjecture, Comment. Math. Helv. 48(1973), 119-135.

 S. Ruscheweyh and V. Singh, On certain extermal problems for functions with positive real part, Proc. Amer. Math. Soc., 61(1976), 329-334.

 R. Singh and S. Singh, Convolution properties of a class of starlike functions, Proc. Amer. Math. Soc., 106(1989), 145-152.

(1) Khalida Inayat Noor and (2) Saqib Hussain

(1) Mathematics Department, COMSATSInstitute of Information Technology, Islamabad, Pakistan

Email: khalidanoor@hotmail.com

(2) Mathematics Department, COMSATS Institute of Information Technology, Abbottabad, Pakistan

Email: saqib_math@yahoo.com
Author: Printer friendly Cite/link Email Feedback Noor, Khalida Inayat; Hussain, Saqib International Journal of Difference Equations Report 9PAKI Jun 1, 2011 3241 Duality for non smooth multiobjective fractional programming under generalized V-p-invexity. A mathematical approach to one dimensional Burgers' equation. Convex functions Operator theory