arXiv subjects

Satwanti Devi

Publications and source records attributed to Satwanti Devi.

5 recordsLinked to original sources

arXiv2014

Inclusion properties of Generalized Integral Transform using Duality Techniques

Let $\mathcal{W}_β^δ(α,γ)$ be the class of normalized analytic functions $f$ defined in the region $|z|<1$ and satisfying \begin{align*} {\rm Re\,} e^{iϕ}\left(\dfrac{}{}(1\!-\!α\!+\!2γ)\!\left({f}/{z}\right)^δ+\left(α\!-\!3γ+γ\left[\dfrac{}{}\left(1-{1}/δ\right)\left({zf'}/{f}\right)+ {1}/δ\left(1+{zf"}/{f'}\right)\right]\right)\right.\\ \left.\dfrac{}{}\left({f}/{z}\right)^δ\!\left({zf'}/{f}\right)-β\right)>0, \end{align*} with the conditions $α\geq 0$, $β<1$, $γ\geq 0$, $δ>0$ and $ϕ\in\mathbb{R}$. For a non-negative and real-valued integrable function $λ(t)$ with $\int_0^1λ(t) dt=1$, the generalized non-linear integral transform is defined as \begin{align*} V_λ^δ(f)(z)= \left(\int_0^1 λ(t) \left({f(tz)}/{t}\right)^δdt\right)^{1/δ}. \end{align*} The main aim of the present work is to find conditions on the related parameters such that $V_λ^δ(f)(z)\in\mathcal{W}_{β_1}^{δ_1}(α_1,γ_1)$, whenever $f\in\mathcal{W}_{β_2}^{δ_2}(α_2,γ_2)$. Further, several interesting applications for specific choices of $λ(t)$ are discussed.

math.CV↗

arXiv2014

Convexity of the Generalized Integral Transform and Duality Techniques

Let $\mathcal{W}_β^δ(α,γ)$ be the class of normalized analytic functions $f$ defined in the domain $|z|<1$ satisfying \begin{align*} {\rm Re\,} e^{iϕ}\left(\dfrac{}{}(1\!-\!α\!+\!2γ)\!\left({f}/{z}\right)^δ+\left(α\!-\!3γ+γ\left[\dfrac{}{}\left(1-{1}/δ\right)\left({zf'}/{f}\right)+ {1}/δ\left(1+{zf''}/{f'}\right)\right]\right)\right.\\ \left.\dfrac{}{}\left({f}/{z}\right)^δ\!\left({zf'}/{f}\right)-β\right)>0, \end{align*} with the conditions $α\geq 0$, $β<1$, $γ\geq 0$, $δ>0$ and $ϕ\in\mathbb{R}$. Moreover, for $0<δ\leq\frac{1}{(1-ζ)}$, $0\leqζ<1$, the class $\mathcal{C}_δ(ζ)$ be the subclass of normalized analytic functions such that \begin{align*} {\rm Re}{\,}\left(1/δ\left(1+zf''/f'\right)+(1-1/δ)\left({zf'}/{f}\right)\right)>ζ,\quad |z|<1. \end{align*} In the present work, the sufficient conditions on $λ(t)$ are investigated, so that the generalized integral transform \begin{align*} V_λ^δ(f)(z)= \left(\int_0^1 λ(t) \left({f(tz)}/{t}\right)^δdt\right)^{1/δ},\quad |z|<1, \end{align*} carries the functions from $\mathcal{W}_β^δ(α,γ)$ into $\mathcal{C}_δ(ζ)$. Several interesting applications are provided for special choices of $λ(t)$.

math.CV↗

arXiv2014

Starlikeness of the generalized integral transform using duality techniques

For $α\geq 0$, $δ>0$, $β<1$ and $γ\geq 0$, the class $\mathcal{W}_β^δ(α,γ)$ consist of analytic and normalized functions $f$ along with the condition \begin{align*} {\rm Re\,} e^{iϕ}(\dfrac{}{}(1\!-\!α\!+\!2γ)\!({f}/{z})^δ+(α\!-\!3γ\!+\!γ[\dfrac{}{}(1-{1}/δ)({zf'}/{f})+ {1}/δ(1+{zf''}/{f'})]).\\ .\dfrac{}{}({f}/{z})^δ\!({zf'}/{f})-β)>0, \end{align*} where $ϕ\in\mathbb{R}$ and $|z|<1$, is taken into consideration. The class $\mathcal{S}^\ast_s(ζ)$ be the subclass of the univalent functions, defined by the analytic characterization ${\rm Re}{\,}({zf'}/{f})>ζ$, for $0\leq ζ< 1$, $0<δ\leq\frac{1}{(1-ζ)}$ and $|z|<1$. The admissible and sufficient conditions on $λ(t)$ are examined, so that the generalized and non-linear integral transforms \begin{align*} V_λ^δ(f)(z)= (\int_0^1 λ(t) ({f(tz)}/{t})^δdt)^{1/δ}, \end{align*} maps the function from $\mathcal{W}_β^δ(α,γ)$ into $\mathcal{S}^\ast_s(ζ)$. Moreover, several interesting applications for specific choices of $λ(t)$ are discussed, that are related to some well-known integral operators.

math.CV↗

arXiv2014

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

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.

math.CV↗

arXiv2013

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

Let $W_β(α,γ)$, $β<1$, 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^{iϕ}\left((1-α+2γ)\frac{f}{z}+(α-2γ)f'+γzf"-β\right)\frac{}{}\right)>0, \quad z\in {\mathbb{D}}, \end{align*} for some $ϕ\in {\mathbb{R}}$ with $α\geq 0$, $γ\geq 0$ and $β< 1$. Let $M(ξ)$, $0\leq ξ\leq 1$, denote the Pascu class of $ξ$-convex functions given by the analytic condition \begin{align*} {\rm Re\,}\frac{ξz(zf'(z))'+(1-ξ)zf'(z)}{ξzf'(z)+(1-ξ)f(z)}>0 \end{align*} which unifies the class of starlike and convex functions. The aim of this paper is to find conditions on $λ(t)$ so that the integral transforms of the form \begin{align*} V_λ(f)(z)= \int_0^1 λ(t) \frac{f(tz)}{t} dt. \end{align*} carry functions from $W_β(α,γ)$ into $M(ξ)$. As applications, for specific values of $λ(t)$, it is found that several known integral operators carry functions from $W_β(α,γ)$ into $M(ξ)$. Results for a more generalized operator related to $V_λ(f)(z)$ are also given.

math.CV↗