On an Integral Transform of a Class of Analytic Functions

and Applied Analysis 3 Let αj j 1, 2, . . . , p and βj j 1, 2, . . . , q be complex numbers with βj / 0,−1,−2, . . . j 1, 2, . . . , q . Then the generalized hypergeometric function pFq is defined by pFq z pFq ( α1, . . . , αp; β1, . . . , βq; z ) ∞ ∑ n 0 α1 n · · · ( αp ) n ( β1 ) n · · · ( βq ) n z n! ( p ≤ q 1), 1.7 where a n is the Pochhammer symbol, defined in terms of the Gamma function, by a n : Γ a n Γ a { 1, n 0, a a 1 · · · a n − 1 , n ∈ N. 1.8 In particular, 2F1 is called the Gaussian hypergeometric function. We note that the pFq series in 1.7 converges absolutely for |z| <∞ if p < q 1 and for z ∈ E if p q 1. We shall also need the following lemma. Lemma 1.1 see 5 . Let β1 < 1, β2 < 1, and η ∈ R. Then, for p, q analytic in E with p 0 q 0 1, the conditions p z > β1 and e q z − β2 > 0 imply e p ∗ q z − δ > 0, where 1 − δ 2 1 − β1 1 − β2 . 2. Main Results We use the notations introduced in 4 . Let μ ≥ 0 and ν ≥ 0 satisfy μ ν α − γ, μν γ. 2.1 When γ 0, then μ is chosen to be 0, in which case, ν α ≥ 0. When α 1 2γ , 2.1 yields μ ν 1 γ 1 μν or μ − 1 1 − ν 0. i For γ > 0, then choosing μ 1 gives ν γ . ii For γ 0, then μ 0 and ν α 1. Theorem 2.1. Let μ ≥ 0, ν ≥ 0 satisfy 2.1 . Further, let δ < 1 be given, and define β β δ, μ, ν by 1 − 1 − δ 2 { 1 − 1 ν ∫1 0 λ t (∫1 0 ds 1 tsμ ) dt ( 1 ν − 1 )∫1 0 λ t (∫∫1 0 dη dζ 1 tηνζμ ) dt }−1 , γ / 0, 1 − 1 − δ 2 { 1 − 1 α ∫1 0 λ t 1 t dt ( 1 α − 1 )∫1 0 λ t (∫1 0 dη 1 tηα ) dt }−1 , γ 0 ( μ 0, ν α > 0 ) . 2.2 If f ∈ Wβ α, γ , then F Vλ f ∈ Wδ 1, 0 ⊂ S. The value of β is sharp. Proof. The case γ 0 μ 0, ν α > 0 corresponds to Theorem 1.5 in 2 . So we assume that γ > 0. 4 Abstract and Applied Analysis Define ( 1 − α 2γ)f z z ( α − 2γ)f ′ z γzf ′′ z H z . 2.3 Writing f z z ∑∞ n 2 anz , it follows that H z 1 ∞ ∑ n 1 an 1 nν 1 ( nμ 1 ) z. 2.4 It is a simple exercise to see that f ′ z H z ∗ 3F2 ( 2, 1 ν , 1 μ ; 1 ν 1 , 1 μ 1 ; z ) . 2.5 Let F z Vλ f z , where Vλ f is defined by 1.2 . Then for γ / 0, we can write F ′ z f ′ z ∗ ∫1 0 λ t 1 − tz H z ∗ 3F2 ( 2, 1 ν , 1 μ ; 1 ν 1 , 1 μ 1 ; z ) ∗ ∫1 0 λ t 1 − tz H z ∗ ∫1 0 λ t 3F2 ( 2, 1 ν , 1 μ ; 1 ν 1 , 1 μ 1 ; tz ) dt. 2.6 Since f ∈ Wβ α, γ , it follows that {eiφ H z − β } > 0 for some φ ∈ R. Now, for each γ > 0, we first claim that [∫1 0 λ t 3F2 ( 2, 1 ν , 1 μ ; 1 ν 1 , 1 μ 1 ; tz ) dt ] > 1 − 1 − δ 2 ( 1 − β) , z ∈ E, 2.7 which, by Lemma 1.1, implies that F ∈ Wδ 1, 0 . Therefore, it suffices to verify the inequality 2.7 . Using the identity which can be checked by comparing the coefficients of z on both sides 3F2 2, b, c;d, e; z d − 1 3F2 1, b, c;d − 1, e; z − d − 2 3F2 1, b, c;d, e; z , 2.8


Introduction
Let A denote the class of analytic functions f defined in the open unit disc E {z : |z| < 1} with the normalizations f 0 f 0 − 1 0, and let S be the subclass of A consisting of functions univalent in E. For any two functions f z z ∞ n 2 a n z n and g z z ∞ n 2 b n z n in A, the Hadamard product or convolution of f and g is the function f * g defined by f * g z z ∞ n 2 a n b n z n . 1.1 For f ∈ A, Fournier and Ruscheweyh 1 introduced the integral operator F z V λ f z : where λ is a nonnegative real-valued integrable function satisfying the condition 1 0 λ t dt 1. This operator contains some well-known operators such as Libera, Bernardi, and Komatu as its special cases. Fournier and Ruscheweyh 1 applied the famous duality theory to show that for a function f in the class Since then, this operator has been studied by a number of authors for various choices of λ t . In another remarkable paper, Barnard et al. in 2 obtained conditions such that V λ f ∈ P 1 β whenever f is in the class with β < 1, γ ≥ 0. Note that for 0 ≤ β < 1, functions in P 1 β ≡ P β satisfy the condition f z > β in E and thus are close-to-convex in E. A domain D in C is close-to-convex if its compliment in C can be written as union of nonintersecting half lines.
In 2008, Ponnusamy and Rønning 3 discussed the univalence of V λ f for the functions in the class In a very recent paper, Ali et al. 4 studied the class where α, γ ≥ 0 and β < 1. In this paper, they obtained sufficient conditions so that the integral transform V λ f maps normalized analytic functions f ∈ W β α, γ into the class of starlike functions. It is evident that In the present paper, we shall mainly tackle the following problems.
To prove one of our results, we shall need the generalized hypergeometric function p F q , so we define it here.
Abstract and Applied Analysis 3 Let α j j 1, 2, . . . , p and β j j 1, 2, . . . , q be complex numbers with β j / 0, −1, −2, . . . j 1, 2, . . . , q . Then the generalized hypergeometric function p F q is defined by where a n is the Pochhammer symbol, defined in terms of the Gamma function, by a n : Γ a n Γ a 1, n 0, a a 1 · · · a n − 1 , n ∈ N. 1.8 In particular, 2 F 1 is called the Gaussian hypergeometric function. We note that the p F q series in 1.7 converges absolutely for |z| < ∞ if p < q 1 and for z ∈ E if p q 1.
We shall also need the following lemma.
Writing f z z ∞ n 2 a n z n , it follows that H z 1 ∞ n 1 a n 1 nν 1 nμ 1 z n .

2.4
It is a simple exercise to see that ; tz dt.

2.6
Since f ∈ W β α, γ , it follows that {e iφ H z − β } > 0 for some φ ∈ R. Now, for each γ > 0, we first claim that which, by Lemma 1.1, implies that F ∈ W δ 1, 0 . Therefore, it suffices to verify the inequality 2.7 . Using the identity which can be checked by comparing the coefficients of z n on both sides 2.9 Abstract and Applied Analysis 5 Thus,

2.10
Therefore, for γ > 0, we have in the view of 2.2 .
To prove the sharpness, let f ∈ W β α, γ be the function determined by Using a series expansion, we see that we can write Then, Abstract and Applied Analysis where ψ n 1 0 λ t t n−1 dt. Equation 2.2 can be restated as

2.17
This shows that the result is sharp.