[Paper Review] Order of Starlikeness and Convexity of certain integral transforms using duality techniques

Satwanti Devi, A. Swaminathan|arXiv (Cornell University)|Jun 25, 2014

Analytic and geometric function theory5 references3 citations

TL;DR

This paper investigates the order of starlikeness and convexity for integral transforms of analytic functions using duality techniques. It derives necessary and sufficient conditions on the kernel function $\lambda(t)$ such that the transform $V_\lambda(f)$ maps functions from the class $\mathcal{W}_\beta(\alpha,\gamma)$ into the Pascu class $M(\sigma,\xi)$, generalizing known results and providing sharper bounds for specific parameter choices, particularly when $\xi=1$ (convexity) and $\sigma=0$.

ABSTRACT

For $α\geq 0$, $β<1$ and $γ\geq 0$, the class $\mathcal{W}_β(α,γ)$ satisfies the condition \begin{align*} { m Re\,} \left( e^{iϕ}\left((1-α+2γ)f/z+(α-2γ)f'+ γzf''-β ight)\frac{}{} ight)>0, \quad ϕ\in {\mathbb{R}},{\,}z\in {\mathbb{D}}; \end{align*} is taken into consideration. The Pascu class of $ξ$-convex functions of order $σ$ $(M(σ,{\,}ξ))$, having analytic characterization \begin{align*} { m Re\,}\frac{ξz(zf'(z))'+(1-ξ)zf'(z)}{ξzf'(z)+(1-ξ)f(z)}>σ,\quad 0\leq σ< 1,\quad z\in {\mathbb{D}}, \end{align*} unifies starlike and convex functions class of order $σ$.The admissible and sufficient conditions on $λ(t)$ are investigated so that the integral transforms \begin{align*} V_λ(f)(z)= \int_0^1 λ(t) \frac{f(tz)}{t} dt, \end{align*} maps the function from $\mathcal{W}_β(α,γ)$ into $M(σ,{\,}ξ)$. Further several interesting applications, for specific choice of $λ(t)$ are discussed which are related to the classical integral transform.

Motivation & Objective

• To determine admissible and sufficient conditions on the kernel $\lambda(t)$ such that the integral transform $V_\lambda(f)$ maps functions from $\mathcal{W}_\beta(\alpha,\gamma)$ into the Pascu class $M(\sigma,\xi)$.
• To generalize and refine existing results on the univalence and convexity properties of integral transforms, particularly for $\xi=0$ (starlikeness) and $\xi=1\,$ (convexity).
• To improve upon earlier bounds for parameters in $\lambda(t)$, especially for the case $\xi=1$, by using duality techniques and integral inequalities.
• To provide new sufficient conditions under which $V_\lambda(f) \in M(\sigma,\xi)$, with explicit constraints on $\sigma$, $\xi$, and the parameters of $\lambda(t)$.

Proposed method

• Utilizes duality techniques to analyze the analytic conditions defining the classes $\mathcal{W}_\beta(\alpha,\gamma)$ and $M(\sigma,\xi)$, linking them through integral transforms.
• Applies the Hadamard product representation to express the transform $V_\lambda(f)(z)$ as a convolution, enabling the use of subordination and differential subordination techniques.
• Derives a key inequality involving $\lambda(t)$, $\mu$, $\nu$, $\xi$, and $\sigma$ by analyzing the logarithmic derivative of $\lambda(t)$ and its second derivative.
• Employs integral representations and logarithmic transformations, particularly $\log(1/t)$, to reduce the problem to verifying non-negativity of quadratic forms in $\log(1/t)$.
• Uses the inequality $\log(1/t) \geq 2(1-t)/(1+t)$ for $t \in (0,1)$ to establish sufficient conditions for the required subordination.
• Considers specific forms of $\lambda(t)$, including power-logarithmic and rational forms, to derive explicit bounds for $\sigma$ and parameter ranges.

Experimental results

Research questions

• RQ1 Under what conditions on $\lambda(t)$ does the integral transform $V_\lambda(f)$ map $\mathcal{W}_\beta(\alpha,\gamma)$ into the Pascu class $M(\sigma,\xi)$?
• RQ2 How do the derived bounds for $\sigma$ and the parameters of $\lambda(t)$ compare to those in prior works, especially for $\xi=1$ (convexity) and $\xi=0$ (starlikeness)?
• RQ3 Can duality techniques yield sharper or more general conditions than existing results for the univalence and order of starlikeness/convexity of $V_\lambda(f)$?
• RQ4 What is the role of the parameters $\mu$, \nu$, $\xi$, and $\sigma$ in determining the admissible class of $\lambda(t)$?
• RQ5 How do specific forms of $\lambda(t)$, such as $t^a \log(1/t)$ or rational functions, affect the resulting bounds on $\sigma$?

Key findings

• For $\lambda(t)$ of the form $t^a \log(1/t)$ with $-1 < a \leq 0$, the transform $V_\lambda(f)$ belongs to $M(\sigma,\xi)$ if $\left(\frac{1}{\xi} + \frac{2}{\mu} - \frac{1}{\nu}\right) \geq 2$ and $0 \leq \sigma \leq \frac{1}{2}\left(\frac{1/\mu - 1/\nu}{1 + 1/\mu - 1/\nu}\right)$.
• When $\xi = 1$, the result improves upon [16, Theorem 5.9], providing a tighter bound for the parameter range of $a$ in $\lambda(t)$.
• For $\lambda(t)$ given by $\frac{(1-k)(3-k)}{2}t^{-k}(1-t^2)$ with $0 \leq k < 1$, the transform maps into $M(\sigma,\xi)$ when $k = 1 - \frac{1}{\xi} - \frac{2}{\mu} + \frac{1}{\nu}$ and $\sigma = \frac{1}{2}$.
• The derived bounds for $\sigma$ are sharper than those in [15, Theorem 5.4] for $\xi = 0$ and [16, Theorem 5.9] for $\xi = 1$, particularly in the range of $a$ and $\sigma$.
• The method successfully generalizes previous results by unifying conditions for both starlikeness ($\xi=0$) and convexity ($\xi=1$) in a single framework using duality.
• The analysis confirms that the class $M(\sigma,\xi)$, which unifies starlike and convex functions, is preserved under the transform $V_\lambda(f)$ under the derived conditions on $\lambda(t)$.