arXiv 1406.64712014-06-25math.CV

利用对偶技巧确定某些积分变换的星形阶和凸性阶

Order of Starlikeness and Convexity of certain integral transforms using duality techniques

Satwanti Devi, A. Swaminathan

备注：: generalization of the previous article

摘要：

对于$α\geq 0$、$β<1$和$γ\geq 0$，考虑满足条件\begin{align*} {\rm Re\,} \left( e^{iϕ}\left((1-α+2γ)f/z+(α-2γ)f'+ γzf''-β\right)\frac{}{}\right)>0, \quad ϕ\in {\mathbb{R}},{\,}z\in {\mathbb{D}}; \end{align*}的类$\mathcal{W}_β(α,γ)$。$ξ$的Pascu类--$σ$阶凸函数$(M(σ,{\,}ξ))$，具有解析特征\begin{align*} {\rm 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*}统一了$σ$阶星形函数类和凸函数类，研究了$λ(t)$的容许条件和充分条件，使得积分变换\begin{align*} V_λ(f)(z)= \int_0^1 λ(t) \frac{f(tz)}{t} dt, \end{align*}将函数从$\mathcal{W}_β(α,γ)$映射到$M(σ,{\,}ξ)$ .进一步讨论了与经典积分变换有关的$λ(t)$的几个有趣的应用。

英文摘要：

For $α\geq 0$, $β<1$ and $γ\geq 0$, the class $\mathcal{W}_β(α,γ)$ satisfies the condition \begin{align*} {\rm Re\,} \left( e^{iϕ}\left((1-α+2γ)f/z+(α-2γ)f'+ γzf''-β\right)\frac{}{}\right)>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*} {\rm 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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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