A Study of the Growth Results for the Hadamard Product of Several Dirichlet Series with Different Growth Indices

Abstract

In this paper, our first purpose is to describe a class of phenomena involving the growth in the Hadamard–Kong product of several Dirichlet series with different growth indices. We prove that (i) the order of the Hadamard–Kong product series is determined by the growth in the Dirichlet series with smaller indices if these Dirichlet series have different growth indices; (ii) the q1-type of the Hadamard–Kong product series is equal to zero if p Dirichlet series are of qj-regular growth, and q1<q2<⋯<qp, qj∈N+, j=1,2,…,p. The second purpose is to reveal the properties of the growth in the Hadamard–Kong product series of two Dirichlet series—when one Dirichlet series is of finite order, the other is of logarithmic order, and two Dirichlet series are of finite logarithmic order—and obtain the growth relationships between the Hadamard–Kong product series and two Dirchlet series concerning the order, the logarithmic order, logarithmic type, etc. Finally, some examples are given to show that our results are best possible.

Higher Transcendental Functions and Their Multi-Disciplinary Applications

“…The geographical distribution of the contributors to this Special Issue is remarkably widely-scattered. Their contributions (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]) originated in many different countries on every continent of the world.…”

“…The subject matter of the first 16 publications (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]) dealt extensively with analytic, univalent, multivalent, and harmonic functions of complex analysis and their quantum or basic (or q-) extensions, the Euler-Poisson-Darboux partial differential equation, approximation theory and associated summability methods, variational inequalities, linear and nonlinear integro-differential equations, growth results involving Dirichlet series, theory and applications of wavelet transforms, analysis of ordinary and partial differentialdifference equations, and several other topics listed in the preceding section.…”