New findings on the periodicity of entire functions and their differential polynomials

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 page

On Relative Convex Sequences

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 page