Related papers: Bohr phenomenon for certain close-to-convex analyt…

For $0<\lambda \leq 1$, let ${\mathcal U}(\lambda)$ denote the family of functions $f(z)=z+\sum_{n=2}^{\infty}a_nz^n$ analytic in the unit disk $\ID$ satisfying the condition $\left |\left (\frac{z}{f(z)}\right )^{2}f'(z)-1\right |<\lambda…

Let $X=\{ X_n\}_{n\in \mathbb{Z}}$ be zero-mean stationary Gaussian sequence of random variables with covariance function $\rho$ satisfying $\rho(0)=1$. Let $\varphi:\mathbb{R}\to\mathbb{R}$ be a function such that…

The logarithmic coefficients $\gamma_n$ of an analytic and univalent function $f$ in the unit disk $\mathbb{D}=\{z\in\mathbb{C}:|z|<1\}$ with the normalization $f(0)=0=f'(0)-1$ is defined by $\log \frac{f(z)}{z}= 2\sum_{n=1}^{\infty}…

In this paper, we introduce a new subclass of close-to-convex harmonic functions. We present a sufficient coefficient condition for a function to be a member of this class. Furthermore, we establish a distortion theorem. These results lay…

In [14,26], new approximation classes of self-referential functions are introduced as fractal versions of the classes of polynomials and rational functions. As a sequel, in the present article, we define a new approximation class consisting…

The purpose of this article is to study Bohr inequalities involving the absolute values of the coefficients of an operator valued function. To be more specific, we establish an operator valued analogue of a classical result regarding the…

For a function $f$ starlike of order $\alpha$, $0\leqslant \alpha <1$, a non-constant polynomial $Q$ of degree $n$ which is non-vanishing in the unit disc $\mathbb{D}$ and $\beta>0$, we consider the function $F:\mathbb{D}\to\mathbb{C}$…

For functions $f(z)=z^p+a_{n+1}z^{p+1}+...$ defined on the open unit disk, the condition $\Re (f'(z)/z^{p-1})>0$ is sufficient for close-to-convexity of $f$. By making use of this result, several sufficient conditions for close-to-convexity…

In this article, Bohr type inequalities for some complex valued harmonic functions defined on the unit disk are given. All the results are sharp.

In 1931 Bohnenblust and Hille proved that for each m-homogeneous polynomial $\sum_{|\alpha| = m} a_\alpha z^\alpha$ on $\C^n$ the $\ell^{\frac{2m}{m+1}}$-norm of its coefficients is bounded from above by a constant $C_m$ (depending only on…

To minimize or upper-bound the value of a function "robustly", we might instead minimize or upper-bound the "epsilon-robust regularization", defined as the map from a point to the maximum value of the function within an epsilon-radius. This…

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 $0<r\le 1$. In 1990, Yamashita conjectured that $\Delta(r,z/f)\le \pi r^2$ for convex…

In this paper we study the problem of testing if an $L_2-$function $f$ belonging to a certain $l_2$-Sobolev-ball $B_t(R)$ of radius $R>0$ with smoothness level $t>0$ indeed exhibits a higher smoothness level $s>t$, that is, belongs to…

For $ -1 \leq B \leq 1$ and $A>B$, let $\mathcal{S}^*[A,B]$ denote the class of generalized Janowski starlike functions consisting of all normalized analytic functions $f$ defined by the subordination $z f'(z)/f(z) \prec (1+ A z)/(1+ B z)$…

We study the $k$-fold symmetric starlike univalent logharmonic mappings of the form $f(z)=zh(z)\overline{g(z)}$ in the unit disk $\mathbb{D}:= \lbrace z \in \mathbb{C}: |z|<1 \rbrace$ with several examples, where $h(z)=\exp…

The well-known Bohr--P\'al theorem asserts that for every continuous real-valued function $f$ on the circle $\mathbb T$ there exists a change of variable, i.e., a homeomorphism $h$ of $\mathbb T$ onto itself, such that the Fourier series of…

This paper examines the coefficient problems for the class of semigroup generators, a topic in complex dynamics that has recently been studied in context of geometric function theory. Further, sharp bounds of coefficient functional such as…

We show that for multivariate Freud-type weights $W_\alpha(x)=\exp(-|x|^\alpha)$, $\alpha>1$, any convex function $f$ on $R^d$ satisfying $fW_\alpha\in L_p(R^d)$ if $1\le p<\infty$, or $\lim_{|x|\to\infty}f(x)W_\alpha(x)=0$ if $p=\infty$,…

The main aim of this article is to establish a sharp improvement of the classical Bohr inequality for bounded holomorphic mappings in the polydisk $\mathbb{D}^n$.We also prove two other sharp versions of the Bohr inequality in the setting…

We consider the averages of a function $ f$ on $ \mathbb R ^{n}$ over spheres of radius $ 0< r< \infty $ given by $ A_{r} f (x) = \int_{\mathbb S ^{n-1}} f (x-r y) \; d \sigma (y)$, where $ \sigma $ is the normalized rotation invariant…