We consider a newly introduced integral operator that depends on an analytic normalized function and generalizes many other previously studied operators. We find the necessary conditions that this ...operator has to meet in order to preserve convex meromorphic functions. We know that convexity has great impact in the industry, linear and non-linear programming problems, and optimization. Some lemmas and remarks helping us to obtain complex functions with positive real parts are also given.
Honoring Lawrence Zalcman’s work, the cover of this volume shows the phase plot of a function that appears in the construction of a special non-normal family of meromorphic functions. Neglecting all ...technicalities, we summarize the basic ideas of this construction and illustrate them by phase plots.
Let
${\mathbb D}$
be the open unit disk, and let
$\mathcal {A}(p)$
be the class of functions f that are holomorphic in
${\mathbb D}\backslash \{p\}$
with a simple pole at
$z=p\in (0,1)$
, and
...$f'(0)\neq 0$
. In this article, we significantly improve lower bounds of the Bloch and the Landau constants for functions in
${\mathcal A}(p)$
which were obtained in Bhowmik and Sen (2023, Monatshefte für Mathematik, 201, 359–373) and conjecture on the exact values of such constants.
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