ﻻ يوجد ملخص باللغة العربية
Let $n$ be a positive integer. Let $mathbf U$ be the unit disk, $pge 1$ and let $h^p(mathbf U)$ be the Hardy space of harmonic functions. Kresin and Mazya in a recent paper found the representation for the function $H_{n,p}(z)$ in the inequality $$|f^{(n)} (z)|leq H_{n,p}(z)|Re(f-mathcal P_l)|_{h^p(mathbf U)}, Re fin h^p(mathbf U), zin mathbf U,$$ where $mathcal P_l$ is a polynomial of degree $lle n-1$. We find or represent the sharp constant $C_{p,n}$ in the inequality $H_{n,p}(z)le frac{C_{p,n}}{(1-|z|^2)^{1/p+n}}$. This extends a recent result of the second author and Markovic, where it was considered the case $n=1$ only. As a corollary, an inequality for the modulus of the $n-{th}$ derivative of an analytic function defined in a complex domain with the bounded real part is obtained. This result improves some recent result of Kresin and Mazya.
In this paper, by making use of a certain family of fractional derivative operators in the complex domain, we introduce and investigate a new subclass $mathcal{P}_{tau,mu}(k,delta,gamma)$ of analytic and univalent functions in the open unit disk $mat
Some sufficient conditions on certain constants which are involved in some first, second and third order differential subordinations associated with certain functions with positive real part like modified Sigmoid function, exponential function and Ja
For an analytic function $f$ defined on the unit disk $|z|<1$, let $Delta(r,f)$ denote the area of the image of the subdisk $|z|<r$ under $f$, where $0<rle 1$. In 1990, Yamashita conjectured that $Delta(r,z/f)le pi r^2$ for convex functions $f$ and i
We introduce the class of analytic functions $$mathcal{F}(psi):= left{fin mathcal{A}: left(frac{zf(z)}{f(z)}-1right) prec psi(z),; psi(0)=0 right},$$ where $psi$ is univalent and establish the growth theorem with some geometric conditions on $psi$ an
We study proper rational maps from the unit disk to balls in higher dimensions. After gathering some known results, we study the moduli space of unitary equivalence classes of polynomial proper maps from the disk to a ball, and we establish a normal