Related papers: On the Logarithmic Coefficients of Close to Convex…
We prove that a general class of measures, which includes $\log$-concave measures, is $\frac{1}{n}$-concave according to the terminology of Borell, with additional assumptions on the measures or on the sets, such as symmetries. This…
The sums of three trigonometric series with logarithmic coefficients are derived by extending an approach first utilized by Lerch. By applying Frullani's theorem to two of these series, two non-trivial integrals involving hyperbolic…
Let $\lambda$ the Barban--Vehov weights, defined in $(1)$. Let $X\ge z_1\ge100$ and $z_2=z_1^\tau$ for some $\tau>1$. We prove that \begin{equation*} \sum_{n\le X}\frac{1}{n}\Bigl(\sum_{\substack{d|n}}\lambda_d\Bigr)^2 \le f(\tau)\frac{\log…
This paper deals with approximation of smooth convex functions $f$ on an interval by convex algebraic polynomials which interpolate $f$ at the endpoints of this interval. We call such estimates "interpolatory". One important corollary of…
In this paper we construct certain type of near-optimal approximations of a class of analytic functions in the unit disc by power series with two distinct coefficients. More precisely, we show that if all the coefficients of the power…
We propose a new approach to deriving quantitative mean field approximations for any probability measure $P$ on $\mathbb{R}^n$ with density proportional to $e^{f(x)}$, for $f$ strongly concave. We bound the mean field approximation for the…
For a periodic function $f$ with bounded variation and integral zero on its period interval, we show that $\sum_{k=1}^\infty c_k^2 (\log\log k)^\gamma <\infty$, $\gamma>4$ implies the almost everywhere convergence of $\sum_{k=1}^\infty c_k…
We investigate the distribution of the logarithmic derivative of the Riemann zeta-function on the line Re(s)=\sigma, where \sigma, lies in a certain range near the critical line \sigma=1/2. For such \sigma, we show that the distribution of…
This paper gives some results for the logarithm of the Riemann zeta-function and its iterated integrals. We obtain a certain explicit approximation formula for these functions. The formula has some applications, which are related with the…
In this paper, we obtain various conditions on the parameters $a,\, b,\, c\,, d$ and $e$ for which the hypergeometric functions $z\,_3F_2(a,b,c;d,e;z)$ to be in the class of all close-to-convex function with respect to some well known…
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…
A starlike function $f$ is characterized by the quantity $zf'(z)/f(z)$ lying in the right half-plane. This paper deals with sharp bounds for certain symmetric Toeplitz determinants whose entries are the coefficients of the functions $f$ for…
In this paper we further study the relationship between convexity and additive growth, building on the work of Schoen and Shkredov (\cite{SS}) to get some improvements to earlier results of Elekes, Nathanson and Ruzsa (\cite{ENR}). In…
In this article, we consider the family of functions $f$ analytic in the unit disk $|z|<1$ with the normalization $f(0)=0=f'(0)-1$ and satisfying the condition $\big |\big (z/f(z)\big )^{2}f'(z)-1\big |<\lambda $ for some $0<\lambda \leq…
In this paper, we find all sums of two Fibonacci numbers which are close to a power of 2. As a corollary, we also determine all Lucas numbers close to a power of 2. The main tools used in this work are lower bounds for linear forms in…
We consider functions of the type $f(z)=z+a_2z^2+a_3z^3+\cdots$ from a family of all analytic and univalent functions in the unit disk. Let $F$ be the inverse function of $f$, given by $F(z)=w+\sum_{n=2}^{\infty}A_nw^n$ defined on some…
In this paper, we first consider the graph of $(F_1,F_{2},\cdots,F_{n})$ on $\overline{\mathbb{D}}^{n},$ where $F_{j}(z)=\bar{z}^{m_{j}}_{j}+R_{j}(z),j=1,2,\cdots,n,$ which has non-isolated CR-singularities if $m_{j}>1$ for some…
In this paper, we determine the sharp estimates for Toeplitz determinants of a subclass of close-to-convex harmonic mappings. Moreover, we obtain an improved version of Bohr's inequalities for a subclass of close-to-convex harmonic…
Various approaches to the numerical representation of the Incomplete Gamma Function F_m(z) for complex arguments z and small integer indexes m are compared with respect to numerical fitness (accuracy and speed). We consider power series,…
Using lattice approximations of Euclidean space, we develop a way to approximate stable processes that are represented by stochastic integrals over Euclidean space. Via a stable version of the Lindeberg-Feller Theorem we show that the…