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Register a new userCertain Estimates of Normalized Analytic Functions
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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.
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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.
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.
In this work, we consider certain class of bi-univalent functions related with shell-like curves related to $kappa-$Fibonacci numbers. Further, we obtain the estimates of initial Taylor-Maclaurin coefficients (second and third coefficients) and Fekete - Szeg{o} inequalities. Also we discuss the special cases of the obtained results.
We introduce the Schur class of functions, discrete analytic on the integer lattice in the complex plane. As a special case, we derive the explicit form of discrete analytic Blaschke factors and solve the related basic interpolation problem.
Via a unified geometric approach, a class of generalized trigonometric functions with two parameters are analytically extended to maximal domains on which they are univalent. Some consequences are deduced concerning commutation with rotation, continuation beyond the domain of univalence, and periodicity.
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