Related papers: Logarithmic convexity of area integral means for a…

We show that the $L^2$ integral mean on $r\D$ of an analytic function in the unit disk $\D$ with respect to the weighted area measure $(1-|z|^2)^\alpha\,dA(z)$, where $-3\le\alpha\le0$, is a logarithmically convex function of $r$ on…

For $0<p<\infty$ and $\alpha\in (-\infty,\infty)$ we determine when the $L^p$ integral mean on $\{z\in\mathbb C: |z|\le r\}$ of an entire function with respect to the Gaussian area measure $e^{-\alpha|z|^2}\,dA(z)$ is logarithmic convex or…

In the present work weighted area integral means $M_{p,\varphi}(f;{\mathrm {Im}}z)$ are studied and it is proved that the function $y\to \log M_{p,\varphi}(f;y)$ is convex in the case when $f$ belongs to a Hardy space on the upper…

This note addresses monotonic growths and logarithmic convexities of the weighted ($(1-t^2)^\alpha dt^2$, $-\infty<\alpha<\infty$, $0<t<1$) integral means $\mathsf{A}_{\alpha,\beta}(f,\cdot)$ and $\mathsf{L}_{\alpha,\beta}(f,\cdot)$ of the…

For $\lambda\ge0$, a $C^2$ function $f$ defined on the unit disk ${{\mathbb D}}$ is said to be $\lambda$-analytic if $D_{\bar{z}}f=0$, where $D_{\bar{z}}$ is the (complex) Dunkl operator given by…

We establish sharp $L^p$ integral mean estimates for $(\alpha,\beta)$-harmonic functions on the unit disk. Explicit bounds for the functions and their partial derivatives are obtained in terms of boundary data, by means of the associated…

For $f$ analytic and close to convex in $D=\{z: |z|< 1\}$, we give sharp estimates for the logarithmic coefficients $\gamma_{n}$ of $f$ defined by $\log \dfrac{f(z)}{z}=2\sum_{n=1}^{\infty} \gamma_{n}z^{n}$ when $n=1, 2,3$.

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…

Let $\mathcal{H}ol(B_d)$ denote the space of holomorphic functions on the unit ball $B_d$ of $\mathbb{C}^d$, $d\ge 1$. Given a log-convex strictly positive weight $w(r)$ on $[0,1)$, we construct a function $f\in\mathcal{H}ol(B_d)$ 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 $f(0)=0=f'(0)-1$ are defined by $\log \frac{f(z)}{z}= 2\sum_{n=1}^{\infty}…

The M\"obius invariant space $\mathcal{Q}_p$, $0<p<\infty$, consists of functions $f$ which are analytic in the open unit disk $\mathbb{D}$ with $$ \|f\|_{\mathcal{Q}_p}=|f(0)|+\sup_{w\in \D} \left(\int_\D |f'(z)|^2(1-|\sigma_w(z)|^2)^p…

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

We show that weighted Bergman projections, corresponding to weights of the form $M(z)(1-|z|^2)^{\alpha}$ where $\alpha>-1$ and $M(z)$ is a radially symmetric, strictly positive and at least $C^2$ function on the unit disc, are $L^p$…

The class $A_\alpha^p$ consists of those analytic functions $f$ in the unit disc such that \[\|f\|_{\alpha,p}^p := |f(0)|^p+\int_0^1 \left(\frac{d}{dr} M_p^p(r,f)\right) (1-r^2)^{\alpha-1} \,dr < \infty,\] where $M_p^p(r,f)$ is the radial…

For $\alpha\ge 0$, let $\mathcal{W}(\alpha)$ be the class of all analytic functions in the unit disk $\mathbb{D}$ with normalization $f(0) = 0 $ and $ f'(0) = 1 $ that satisfy the relation $Re\,\{f'(z) + \alpha z f''(z)\} > 0$. This article…

We investigate $g$-functions and Lusin's area type integrals related to certain multi-dimensional Dunkl and Laguerre settings. We prove that the considered square functions are bounded on weighted $L^p$, $1<p<\infty$, and from $L^1$ into…

We consider the space $\bk^1_{\log^\alpha}$, of analytic functions on the unit disk $\D,$ defined by the requirement $\int_\D|f'(z)|\phi(|z|)\,dA(z)<\infty,$ where $\phi(r)=\log^\alpha(1/(1-r))$ and show that it is a predual of the…

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}…

Let $L_H$ denote the set of all normalized locally one-to-one and sense-preserving harmonic functions in the unit disc $\Delta$. It is well-known that every complex-valued harmonic function in the unit disc $\Delta$ can be uniquely…

The purpose of this paper is to introduce the logarithmic mean of two convex functionals that extends the logarithmic mean of two positive operators. Some inequalities involving this functional mean are discussed as well. The operator…