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Register a new userSharp Bohr Radius Constants For Certain Analytic Functions
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The Bohr radius for a class $mathcal{G}$ consisting of analytic functions $f(z)=sum_{n=0}^{infty}a_nz^n$ in unit disc $mathbb{D}={zinmathbb{C}:|z|<1}$ is the largest $r^*$ such that every function $f$ in the class $mathcal{G}$ satisfies the inequality begin{equation*} dleft(sum_{n=0}^{infty}|a_nz^n|, |f(0)|right) = sum_{n=1}^{infty}|a_nz^n|leq d(f(0), partial f(mathbb{D})) end{equation*} for all $|z|=r leq r^*$, where $d$ is the Euclidean distance. In this paper, our aim is to determine the Bohr radius for the classes of analytic functions $f$ satisfying differential subordination relations $zf(z)/f(z) prec h(z)$ and $f(z)+beta z f(z)+gamma z^2 f(z)prec h(z)$, where $h$ is the Janowski function. Analogous results are obtained for the classes of $alpha$-convex functions and typically real functions, respectively. All obtained results are sharp.
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This paper mainly uses the nonnegative continuous function ${zeta_n(r)}_{n=0}^{infty}$ to redefine the Bohr radius for the class of analytic functions satisfying $real f(z)<1$ in the unit disk $|z|<1$ and redefine the Bohr radius of the alternating series $A_f(r)$ with analytic functions $f$ of the form $f(z)=sum_{n=0}^{infty}a_{pn+m}z^{pn+m}$ in $|z|<1$. In the latter case, one can also get information about Bohr radius for even and odd analytic functions. Moreover, the relationships between the majorant series $M_f(r)$ and the odd and even bits of $f(z)$ are also established. We will prove that most of results are sharp.
Let $phi$ be a normalized convex function defined on open unit disk $mathbb{D}$. For a unified class of normalized analytic functions which satisfy the second order differential subordination $f(z)+ alpha z f(z) prec phi(z)$ for all $zin mathbb{D}$, we investigate the distortion theorem and growth theorem. Further, the bounds on initial logarithmic coefficients, inverse coefficient and the second Hankel determinant involving the inverse coefficients are examined.
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 expansion/contraction properties of some common classes of mappings of the Euclidean space ${mathbb R}^n, nge 2,,$ with respect to the distance ratio metric. The first main case is the behavior of Mobius transformations of the unit ball in ${mathbb R}^n$ onto itself. In the second main case we study the polynomials of the unit disk onto a subdomain of the complex plane. In both cases sharp Lipschitz constants are obtained.
For $frac12<p<infty$, $0<q<infty$ and a certain two-sided doubling weight $omega$, we characterize those inner functions $Theta$ for which $$|Theta|_{A^{p,q}_omega}^q=int_0^1 left(int_0^{2pi} |Theta(re^{itheta})|^p dthetaright)^{q/p} omega(r),dr<infty.$$ Then we show a modified version of this result for $pge q$. Moreover, two additional characterizations for inner functions whose derivative belongs to the Bergman space $A_omega^{p,p}$ are given.
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