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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 [5].

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 [2].

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 [7].

Lemma 2.3.[6]. 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.[8]. 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 [4].

(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 [4].

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

[1] 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.

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

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

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

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

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

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

[8] 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
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Date:Jun 1, 2011
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