On the Convexity of the Bernardi Integral Operator

TL;DR

The paper analyzes the convexity properties of the Bernardi integral operator ${\mathcal B}_c$ on starlike classes and establishes a unified framework for preservation results. It proves that for $0<M< M_c$ with $M_c=\sqrt{c^2+1}-c$, ${\mathcal B}_c$ maps the $S_M$-class into the convex class $K$, with the Libera case ($c=1$) yielding convexity for $0<M<\sqrt{2}-1$ and improving bounds on the Libera constant $M_{\mathcal L}$. A general subordination-based method shows ${\mathcal B}_c({\mathcal F})\subset{\mathcal F}$ for convex $\Psi$ and $\mathcal F={f: \frac{zf'(z)}{f(z)}\prec \Psi(z)}$, unifying many known and new preservation results, including Ma-Minda starlike classes and Janowski-type families. The work further extends to nonlinear and $p$-valent transforms, outlining a broad invariant-subclass framework and discussing implications for spirallike and other generalized function classes.

Abstract

We prove that the Bernardi Integral Operator maps certain classes of bounded starlike functions into the class of convex functions, improving the result of Oros and Oros. We also present a general unified method for investigating various other integral operators that preserve many of the previously studied subclasses of univalent and p-valent functions.

Theorems & Definitions (19)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• proof
• proof