Integral transforms of functions to be in the Pascu class using duality techniques

Let $W_{\beta}(\alpha,\gamma)$, $\beta<1$, denote the class of all normalized analytic functions $f$ in the unit disc ${\mathbb{D}}=\{z\in {\mathbb{C}}: |z|<1\}$ such that \begin{align*} {\rm Re\,} \left(e^{i\phi}\left((1-\alpha+2\gamma)\frac{f}{z}+(\alpha-2\gamma)f'+\gamma zf"-\beta\right)\frac{}{}\right)>0, \quad z\in {\mathbb{D}}, \end{align*} for some $\phi\in {\mathbb{R}}$ with $\alpha\geq 0$, $\gamma\geq 0$ and $\beta<1$. Let $M(\xi)$, $0\leq \xi\leq 1$, denote the Pascu class of $\xi$-convex functions given by the analytic condition \begin{align*} {\rm Re\,}\frac{\xi z(zf'(z))'+(1-\xi)zf'(z)}{\xi zf'(z)+(1-\xi)f(z)}>0 \end{align*} which unifies the class of starlike and convex functions. The aim of this paper is to find conditions on $\lambda(t)$ so that the integral transforms of the form \begin{align*} V_{\lambda}(f)(z)= \int_0^1 \lambda(t) \frac{f(tz)}{t} dt. \end{align*} carry functions from $W_{\beta}(\alpha,\gamma)$ into $M(\xi)$. As applications, for specific values of $\lambda(t)$, it is found that several known integral operators carry functions from $W_{\beta}(\alpha,\gamma)$ into $M(\xi)$. Results for a more generalized operator related to $V_\lambda(f)(z)$ are also given.


Introduction
Let A denote the class of all functions f analytic in the open unit disc D = {z ∈ C : |z| < 1} with the normalization f (0) = f ′ (0) − 1 = 0 and S be the class of functions f ∈ A that are univalent in D. A function f ∈ S is said to be starlike (S * ) or convex (C), if f maps D conformally onto the domains, respectively, starlike with respect to origin and convex. Note that in D, if f ∈ C ⇐⇒ zf ′ ∈ S * follows from the well-known Alexander theorem (see [5] for details). An useful generalization of the class S * is the class S * (σ) that has the analytic characterisation S * (σ) = f ∈ A : Re zf ′ f > σ; 0 ≤ σ < 1 and S * (0) ≡ S * . Various generalization of classes S * and C are abundant in the literature. One such generalization is the following: A function f ∈ A is said to be in the Pascu class of α -convex functions of order σ if [8] Re or in other words This class is denoted by M(α, σ). Even though, this class is known as Pascu class of α -convex functions of order σ, since we use the parameter α for another important class, we denote this class by M(ξ, σ), 0 ≤ ξ ≤ 1, and we remark that, in the sequel, we only consider the class M(ξ) := M(ξ, 0). Clearly M(0) = S * and M(1) = C which implies that this class M(ξ) is a smooth passage between the class of starlike and convex functions. The main objective of this work is to find conditions on the non-negative real valued integrable function λ(t) satisfying 1 0 λ(t)dt = 1, such that the operator is in the class M(ξ). Note that this operator was introduced in [6]. To investigate this admissibility property the class to which the function f belongs is important. Let W β (α, γ), α ≥ 0, γ ≥ 0 and β < 1, denote the class of all normalized analytic functions f in the unit disc D such that for some φ ∈ R. This class and its particular cases were considered by many authors so that the corresponding operator given by (1.1) is univalent and in M(ξ) for some particular values of α, β, γ and ξ. This work was motivated in [6] by studying the conditions under which V λ (W β (1, 0)) ⊂ M(0) and generalized in [7] by studying the case V λ (W β (α, 0)) ⊂ M(0). Similar situation for the convex case, namely V λ (W β (1, 0)) ⊂ M(1) was initiated in [4]. After several generalizations by many authors, recently, the conditions under which V λ (W β (α, γ)) ⊂ M(0) was obtained in [1] and the corresponding results for the convex case so that V λ (W β (α, γ)) ⊂ M(1) was obtained in [3]. Applications involving several well known integral transforms were studied in [1] and [3] (see also [11]) For all the literature involving the complete study in this direction so far we refer to [1,3,9,11] and references therein.
In this work, we find conditions on λ(t) so that V λ (W β (α, γ)) ⊂ M(ξ) using duality techniques which are presented in Section 2. As applications, in Section 3, we consider particular values for λ(t) in (1.1) so that results for some of the well-known integral operators can be deduced. A more generalized operator introduced in [4] is considered in Section 4 for similar type of results.
Note 1. Since the case γ = 0 is considered in [9], we only consider results for the case γ > 0, except for Theorem 3.2 (see Remark 3.2).
Next we introduce two known auxiliary functions [1]. Let and Here φ −1 µ,ν denotes the convolution inverse of φ µ,ν such that φ µ,ν * φ −1 µ,ν = 1/(1 − z). By * , we mean the following: If f and g are in A with the power series expansions f (z) = ∞ k=0 a k z k and g(z) = ∞ k=0 b k z k respectively, then the convolution or Hadamard product of f and g is given by h(z) = ∞ k=0 a k b k z k . Since ν ≥ 0, µ ≥ 0, when γ ≥ 0, making the change of variables u = t ν , v = s µ in (1.4) result in writing ψ µ,ν as Now let g be the solution of the initial value-problem satisfying g(0) = 1. The series solution is given by Let q be the solution of the differential equation (1.7) satisfying q(0) = 0. The series solution of q(t) is given by Note that q(t) also satisfies 2q(t) = tg ′ (t) + g(t) + 1.
Our main results is the generalization of the following results given in [1] and [3]. The necessary and sufficient conditions under which the operator V λ (f (z)) carries the function f (z) from W β (α, γ), to the classes S * and C, respectively are given in next two results.

Main Results
We start with a result that gives both necessary and sufficient condition for an integral transform that satisfies the admissibility property of the class W β (α, γ), which contain non-univalent functions also, to the Pascu class M(ξ).
where g(t) and q(t) are defined by the differential equations given in (1.5) and (1.7) respectively. Assume that The value of β is sharp.
To verify sharpness, let W β (α, γ) be the solution of the differential equation where β < β 0 satisfies (2.1). Further simplification using (1.6) and (1.8) gives This means Using (2.4), a simple computation gives This means Hence zK ′ (z) = 0 for some z ∈ D, so K(z) is not even locally univalent in D. This shows that the result is sharp for β.

Proof. Consider
Integration by parts gives by a simple computation. It is easy to see that, from Theorem (1.3) and (1.4) is decreasing on (0, 1) which is nothing but (2.5) and the proof is complete by applying Theorem 2.1.
Remark 2.2. Even though, we did not consider the case γ = 0, even at γ = 0, Theorem 2.2 does not reduces to a similar result given in [9]. This is due to the fact that, our condition (2.5) has the term (1 − t 2 ) in the denominator, whereas the corresponding result in [9] has the term log(1/t) and hence has different condition.

Applications
It is difficult to check the condition given in Section 2, for V λ (W β (α, γ)) ⊂ M(ξ). In order to find applications, simplified conditions are required. For this purpose, from (2.5), it is enough to show that is increasing on (0,1) which is equivalent of having is decreasing on (0,1), where Λ ν (t) and Π µ, ν (t) are defined in (1.11) and (1.12). It is enough to have g ′ (t) ≤ 0. Let . So to satisfy the above condition we need to have Since Λ ν (1) = 0 and Π µ, ν (1) = 0 we get L(1) = 0. This implies that it suffies to have L(t) is increasing on (0,1), which means This inequality can further be reduced to is true. So, in order to obtain further results, we check conditions (3.2) and (3.3). Note that, whenever λ(1) = λ ′ (1) = 0, from (3.1) we see that it is sufficient to check (3.2) as there is no necessity for the condition given by (3.3).

So the inequality (3.3) is satisfied. If the inequality (3.2) holds then
On further simplification this inequality reduces to Clearly (c + 1) 2 > 0. Hence using c 2 > −(2c + 1) and substituting in (3.5), we get which is true by the hypothesis.
For the case γ = 0, the integral operator is decreasing on (0, 1), which can be easily obtained as in Theorem 2.2. Let , where for the integral operator to be in Substituting the value of p(t) and p ′ (t) in the above equation, we have J(t) Consider the case when λ(t) = (c + 1)t c , where c > −1. So the inequality (3.7) on simplification reduces to If c > 1 + 1 α and α ≤ ξ, then for t = 0, (3.8) is satisfied.
Since α ≤ ξ, we get satisfying the hypothesis of theorem which gives V λ (f )(z) given by (3.6) is in M(ξ).
Case(i) a < b: Since a > −1 and b > −1, so (a + 1)(b + 1) (b − a) > 0. we need only to show Hence A(t) > 0 for all t ∈ (0, 1). Now for B(t) to be positive, it is enough to show that Case(ii) b < a: Since a > −1 and b > −1, so (a + 1)(b + 1) (b − a) < 0. We need only to show for B(t) to be positive, it is enough to show that which is satisfied by the given condition on a. Case(iii) a = b ≤ 0: Changing inequality (3.2), which is true for a = 0. Hence we only consider the situation a < 0. Substituting the values of λ(t), λ ′ (t) and λ ′′ (t) in (3.2), an easy computation shows that for 0 ≤ ξ ≤ 1, it suffices to show that the expressions are non-negative. Since log 1 t is positive, the non-negativity of the first expression follows from hypothesis (iii) of the theorem. Similar observation shows that the second expression reduces to a(a − 1) − a µ ≥ 0 and a µ − 2a + 1 ≥ 0 using log 1 t is positive. These two inequalities, for a < 0, gives which is hypothesis (iii). The proof is complete.
, we take C − A − B = p − 1 and B = c + 1 so that λ(t) takes the form We complete the proof by applying Theorem (3.4) and using a simple computation to obtain c < min{0, 1 + 1 µ } = 0.

A generalized integral operator
In this section for the functions f ∈ W β (α, γ), we consider another integral operator introduced in [4] and find the admissibility conditions to be in the class M(ξ).