Related papers: On the Logarithmic Coefficients of Close to Convex…

Let $U\subseteq\mathbb{R}^{n}$ be open and convex. We show that every (not necessarily Lipschitz or strongly) convex function $f:U\to\mathbb{R}$ can be approximated by real analytic convex functions, uniformly on all of $U$. In doing so we…

We establish an asymptotic formula for the logarithmic mean value of a 1-bounded multiplicative function that is sharp in many cases of interest. We derive from it a variety of applications, making progress on several old problems. As a…

We show that $C^0$-fine approximation of convex functions by smooth (or real analytic) convex functions on $\R^d$ is possible in general if and only if $d=1$. Nevertheless, for $d\geq 2$ we give a characterization of the class of convex…

Let ${\mathcal A}$ be the class of functions $f$ that are analytic in the unit disk ${\mathbb D}$ and normalized such that $f(z)=z+a_2z^2+a_3z^3+\cdots$. Let $0<\lambda\le1$ and \[ {\mathcal U}(\lambda) = \left\{ f\in{\mathcal A}: \left…

Let function $f$ be normalized, analytic and univalent in the unit disk ${\mathbb D}=\{z:|z|<1\}$ and $f(z)=z+\sum_{n=2}^{\infty} a_n z^n$. Using a method based on Grusky coefficients we study several problems over that class of univalent…

We prove various theorems on approximation using polynomials with integer coefficients in the Bernstein basis of any given order. In the extreme, we draw the coefficients from $\{ \pm 1\}$ only. A basic case of our results states that for…

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…

For functions $f(z)= z+ a_2 z^2 + a_3 z^3 + \cdots$ in various subclasses of normalized analytic functions, we consider the problem of estimating the generalized Zalcman coefficient functional $\phi(f,n,m;\lambda):=|\lambda a_n a_m…

Let $\alpha>0$ be a constant, let $\ell\ge0$ be an integer, and let $\Gamma(z)$ denote the classical Euler gamma function. With the help of the integral representation for the Riemann zeta function $\zeta(z)$, by virtue of a monotonicity…

In the $1970$s Nicolas proved that the coefficients $p_d(n)$ defined by the generating function \begin{equation*} \sum_{n=0}^{\infty} p_d(n) \, q^n = \prod_{n=1}^{\infty} \left( 1- q^n\right)^{-n^{d-1}} \end{equation*} are log-concave for…

For $-1\le B<A\le 1$, let $\mathcal{S}^*(A,B)$ denote the class of normalized analytic functions $f(z)= z+\sum_{n=2}^{\infty}a_n z^n$ in $|z|<1$ which satisfy the subordination relation $zf'(z)/f(z)\prec (1+Az)/(1+Bz)$ and $\Sigma^*(A,B)$…

The authors consider the class $\F$ of normalized functions $f$ analytic in the unit disk $\ID$ and satisfying the condition $${\rm Re}\left(1+\frac{zf''(z)}{f'(z)}\right)>-\frac{1}{2},\quad z\in\D. $$ Recently, Ponnusamy et al.…

Let $\mathcal{H}$ denote the class of harmonic functions $f$ in $\mathbb{D}:= \{z\in \mathbb{C}:|z| < 1\}$ normalized by $f(0) = 0 = f_z(0) -1$. For $\alpha \geq 0$, we consider the following class $$\mathcal{W}^0_{\mathcal{H}}(\alpha):=…

In this article, we investigate the extremal properties of logarithmic coefficients for the class $\mathcal{S}_{ch}^*$ of starlike functions associated with the hyperbolic cosine function. We establish the sharp upper bounds for the initial…

We study approximation of Boolean functions by low-degree polynomials over the ring $\mathbb{Z}/2^k\mathbb{Z}$. More precisely, given a Boolean function $F:\{0,1\}^n \rightarrow \{0,1\}$, define its $k$-lift to be $F_k:\{0,1\}^n \rightarrow…

The Hankel determinant $H_{2,1}(F_{f}/2)$ is defined as: \begin{align*} H_{2,1}(F_{f}/2):= \begin{vmatrix} \gamma_1 & \gamma_2 \gamma_2 & \gamma_3 \end{vmatrix}, \end{align*} where $\gamma_1, \gamma_2,$ and $\gamma_3$ are the first, second…

Let $\mathcal{A}$ denote the class of analytic functions $f$ on the unit disc $\mathbb{D}=\{z\in\mathbb{C}:\;|z|<1\}$ normalized by $f(0)=0$ and $f^{\prime}(0)=1$. In the present article, we consider and $\mathcal{F}(c)$ the subclasses of…

Let $f_{\bf c}(r)=\sum_{n=0}^\infty e^{c_n}r^n$ be an analytic function; ${\bf c}=(c_n)\in l_\infty$. We assume that $r$ is some logarithmically convex and lower semicontinuous functional on a locally convex topological space $L$. In this…

The primary objective of this paper is to derive sharp bounds for the norms of the Schwarzian and pre-Schwarzian derivatives in the Ozaki close-to-convex functions $f$, expressed in terms of their value $f^{\prime\prime}(0)$, in particular,…

Logarithmic coefficients play a crucial role in the theory of univalent functions. In this study,we focus on the classes $\mathcal{S}_e^\ast$ and $\mathcal{C}_e$ of starlike and convex functions, respectively, \begin{align*}…