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Register a new userOn real part theorem for the higher derivatives of analytic functions in the unit disk
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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.
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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 $mathbb{U}$. In particular, for functions in the class $mathcal{P}_{tau,mu}(k,delta,gamma)$, we derive sufficient coefficient inequalities, distortion theorems involving the above-mentioned fractional derivative operators, and the radii of starlikeness and convexity. In addition, some applications of functions in the class $mathcal{P}_{tau,mu}(k,delta,gamma)$ are also pointed out.
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 Janowski function are obtained so that the analytic function p normalized by the condition p(0) = 1, is subordinate to Janowski function. The admissibility conditions for Janowski function are used as a tool in the proof of the results. As application, several sufficient conditions are also computed for Janowski starlikeness.
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 it was finally settled in 2013 by Obradovi{c} and et. al.. In this paper, we consider a class of analytic functions in the unit disk satisfying the subordination relation $zf(z)/f(z)prec (1+(1-2beta)alpha z)/(1-alpha z)$ for $0le beta<1$ and $0<alphale 1$. We prove Yamashitas conjecture problem for functions in this class, which solves a partial solution to an open problem posed by Ponnusamy and Wirths.
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 $psi$ and obtain the Koebe domain with some related sharp inequalities. Note that functions in this class may not be univalent. As an application, we obtain the growth theorem for the complete range of $alpha$ and $beta$ for the functions in the classes $mathcal{BS}(alpha):= {fin mathcal{A} : ({zf(z)}/{f(z)})-1 prec {z}/{(1-alpha z^2)},; alphain [0,1) }$ and $mathcal{S}_{cs}(beta):= {fin mathcal{A} : ({zf(z)}/{f(z)})-1 prec {z}/({(1-z)(1+beta z)}),; betain [0,1) }$, respectively which improves the earlier known bounds. The sharp Bohr-radii for the classes $S(mathcal{BS}(alpha))$ and $mathcal{BS}(alpha)$ are also obtained. A few examples as well as certain newly defined classes on the basis of geometry are also discussed.
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 form for these equivalence classes. We also prove that all rational proper maps from the disk to a ball are homotopic in target dimension at least $2$.
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