Related papers: On Log-concavity of the Generalized Marcum Q Funct…

Some subadditivity results involving symmetric (unitarily invariant) norms are obtained. For instance, if $g(t)=\sum_{k=0}^m a_kt^k$ is a polynomial of degree $m$ with non-negative coefficients, then, for all positive operators $A,\,B$ and…

Two conjectures of Su and Wang (2008) concerning binomial coefficients are proved. For $n\geq k\geq 0$ and $b>a>0$, we show that the finite sequence $C_j=\binom{n+ja}{k+jb}$ is a P\'{o}lya frequency sequence. For $n\geq k\geq 0$ and…

Let $\mathcal{S}$ denote the class of analytic and univalent ({\it i.e.}, one-to-one) functions $ f(z)= z+\sum_{n=2}^{\infty}a_n z^n$ in the unit disk $\mathbb{D}=\{z\in \mathbb{C}:|z|<1\}$. For $f\in \mathcal{S}$, In 1999, Ma proposed the…

We consider the partial theta function $\theta (q,z):=\sum _{j=0}^{\infty}q^{j(j+1)/2}z^j$, where $(q,z)\in \mathbb{C}^2$, $|q|<1$. We show that for any $0<\delta _0<\delta <1$, there exists $n_0\in \mathbb{N}$ such that for any $q$ with…

We consider the partial theta function $\theta (q,z):=\sum _{j=0}^{\infty}q^{j(j+1)/2}z^j$, where $z\in \mathbb{C}$ is a variable and $q\in \mathbb{C}$, $0<|q|<1$, is a parameter. Set $D(a):=\{ q\in \mathbb{C}$, $0<|q|\leq a$, $\arg (q)\in…

It is proved that for any non-empty finite subset $Q$ of the square numbers, $ |Q+Q|\geq C'|Q|(\log |Q|)^{1/3+o(1)} $. This result essentially is proved -- with the same tools -- by Mei-Chu Chang. See in J. Funct. Anal. 207 (2004), no 2,…

We study the dimensional Brunn-Minkowski inequality for even log-concave probability measures $\mu$ on $\mathbb{R}^n$ via an analytic approach based on diffusion operators and gradient estimates. Our main result asserts that for every pair…

We prove log-concavity for the function counting partitions without sequences. We use an exact formula for a mixed-mock modular form of weight zero, explicit estimates on modified Kloosterman sums and analytic techniques. Finally, we…

We consider certain subfamilies, of the family of univalent functions in the open unit disk, defined by means of sufficient coefficient conditions for univalency. This article is devoted to studying the problem of the well-known conjecture…

We prove new, general versions of Bernstein-von Mises theorem for both well-specified and misspecified models when the log-likelihood is concave in the parameter and the prior distribution is log-concave. Unlike classical versions of…

We show that the porous medium equation does not in general preserve $\alpha$-concavity of the pressure for $0\le\alpha<1/2$ or $1/2<\alpha\le 1$. In particular, this resolves an open problem of V\'azquez on whether concavity of pressure is…

Let $f: \mathbb{N}^2 \mapsto \mathbb{C}$ be an arithmetic function of two variables. We study the existence of the limit: \[\displaystyle \lim_{x \to \infty} \frac{1}{x^2 (\log x)^{k-1}} \sum_{n_1 , n_2 \le x} f (n_1, n_2) \] where $k$ is a…

In 1987, Alzer posed a conjecture on generalized logarithmic mean, which was introduced by Stolarsky in 1975. To prove Alzer's conjecture, Lou posed a conjecture on generalized inverse harmonic mean in 1995. By proving Lou's conjecture, the…

For every $q\in(0,1)$ and $0\le \alpha<1$ we define a class of analytic functions, the so-called $q$-starlike functions of order $\alpha$, on the open unit disk. We study this class of functions and explore some inclusion properties with…

Two transforms of functions on a half-line are considered. It is proved that their composition gives a concave majorant for every nonnegative function. In particular, this composition is the identity transform on the class of nonnegative…

A generalised Struve function has recently been introduced by Ali, Mondal and Nisar [J. Korean Math. Soc. {\bf 54} (2017) 575--598] as \[(\frac{1}{2} z)^{\nu+1}\sum_{n=0}^\infty\frac{(\frac{1}{2} z)^{2n}}{\Gamma(n+\frac{3}{2})…

In 1993 one of the authors formulated some conjectures on monotonicity of ratios for exponential series sections. They lead to more general conjecture on monotonicity of ratios of Kummer hypergeometric functions and was not proved from…

We show that for every mean zero log-concave real random variable $X$ one has $\|X\|_p \leq \frac{p}{q} \|X\|_q$ for $p \geq q \geq 1$, going beyond the well-known case of symmetric random variables. We also prove that in the class of…

A conjecture posed by Chalmoukis in 2020 states that if $T_{g,a}:H^p\to H^q(0<q<p<\infty)$ is bounded, then $g$ must be in $H^{\frac{pq}{p-q}}$. In this article, we provide a positive answer to the aforementioned conjecture. We also…

The normalized incomplete beta function can be defined either as cumulative distribution function of beta density or as the Gauss hypergeometric function with one of the upper parameters equal to unity. Logarithmic concavity/convexity of…