Search

Given a non-negative real sequence $\{c_n\}_n$ such that the series $\sum_{n=1}^{\infty}c_n$ diverges, it is known that the size of an infinite subset $A\subset\mathbb{N}$ can be measured in terms of the linear density such that the sub-series $\sum_{n\in A}c_n$ either (a) converges or (b) still diverges. The purpose of this research is to study these convergence/divergence questions by measuring the size of the set $A\subset\mathbb{N}$ in a more precise way in terms of the recently introduced asymptotic $\psi$-density. The convergence of the associated sub-signed series $\sum_{n=1 }^{\infty}m_nc_n$ is also discussed, where $\{m_n\}_n$ is a real sequence with values restricted to the set $\{-1, 0, 1\}$., Comment: 27 pages

The sizes of subsets of the natural numbers are typically quantified in terms of asymptotic (linear) and logarithmic densities. These concepts have been generalized to weighted $w$-densities, where a specific weight function $w$ plays a key role. In this paper, a parallel theory of asymptotic $\psi$-densities is introduced, where the weight is expressed slightly differently in terms of differentiable functions $\psi$, which are either concave or convex and satisfy certain asymptotic properties. Alternative new proofs for known results on analytic and Abel densities are also given., Comment: 27 pages

In this paper, we introduce the concept of relative convex sequences and establish their fundamental properties, highlighting their similarities to those of convex sequences. Additionally, we prove new inequalities of the Lupas and Hermite-Hadamard-Fej\'er type for relative convex sequences. In certain cases, and as an application, we show how the concept of relative convexity can facilitate the derivation of new inequalities for convex sequences., Comment: 12 pages

In this paper, we seek to explore under what conditions the periodicity of an entire function $ f(z) $ follows from the periodicity of a differential polynomial in $ f(z) $. We improve and generalize some earlier results and we give other new findings., Comment: 15 pages

A theorem of A. and C. R\'enyi on periodic entire functions states that an entire function $f(z) $ must be periodic if $ P(f(z)) $ is periodic, where $ P(z) $ is a non-constant polynomial. By extending this theorem, we can answer some open questions related to the conjecture of C. C. Yang concerning periodicity of entire functions. Moreover, we give more general forms for this conjecture and we prove, in particular, that $ f(z) $ is periodic if either $ P(f(z)) f^{(k)}(z) $ or $ P(f(z))/f^{(k)}(z) $ is periodic, provided that $ f(z) $ has a finite Picard exceptional value. We also investigate the periodicity of $ f(z) $ when $ f(z)^{n}+a_{1} f^{\prime}(z)+\cdots+a_{k} f^{(k)}(z) $ is periodic. In all our results, the possibilities for the period of $ f(z) $ are determined precisely.

Meromorphic solutions of non-linear differential equations of the form $f^n+P(z,f)=h$ are investigated, where $n\geq 2$ is an integer, $h$ is a meromorphic function, and $P(z,f)$ is differential polynomial in $f$ and its derivatives with small functions as its coefficients. In the existing literature this equation has been studied in the case when $h$ has the particular form $h(z)=p_1(z)e^{\alpha_1(z)}+p_2(z)e^{\alpha_2(z)}$, where $p_1, p_2$ are small functions of $f$ and $\alpha_1,\alpha_2$ are entire functions. In such a case the order of $h$ is either a positive integer or equal to infinity. In this article it is assumed that $h$ is a meromorphic solution of the linear differential equation $h'' +r_1(z) h' +r_0(z) h=r_2(z)$ with rational coefficients $r_0,r_1,r_2$, and hence the order of $h$ is a rational number. Recent results by Liao-Yang-Zhang (2013) and Liao (2015) follow as special cases of the main results., Comment: 29 pages

In this paper, we study the value distribution of zeros of certain nonlinear difference polynomials of entire functions of finite order., Comment: 17 pages

In this paper, we study uniqueness problems for an entire function that shares small functions of finite order with their difference operators. In particular, we give a generalization of results in [2,3,13]., Comment: 19 pages

We investigate uniqueness problems for an entire function that shares two small functions of finite order with their difference operators. In particular, we give a generalization of a result in $[2]$., Comment: 14 pages, accepted for publication