Pseudostarlike and pseudoconvex solutions of a differential equation with exponential coefficients

Abstract

Dirichlet series $F(s)=e^{s}+\sum_{k=1}^{\infty}f_ke^{s\lambda_k}$ with the exponents $1<\lambda_k\uparrow+\infty$ and the abscissa of absolute convergence $\sigma_a[F]\ge 0$ is said to be pseudostarlike of order $\alpha\in [0,\,1)$ and type $\beta \in (0,\,1]$ if$\left|\dfrac{F'(s)}{F(s)}-1\right|<\beta\left|\dfrac{F'(s)}{F(s)}-(2\alpha-1)\right|$\ for all\ $s\in \Pi_0=\{s\colon \,\text{Re}\,s<0\}$. Similarly, the function $F$ is said to be pseudoconvex of order $\alpha\in [0,\,1)$ and type $\beta \in (0,\,1]$ if$\left|\dfrac{F''(s)}{F'(s)}-1\right|<\beta\left|\dfrac{F''(s)}{F'(s)}-(2\alpha-1)\right|$\ for all\ $s\in \Pi_0$. Some conditions are found on the parameters $b_0,\,b_1,\,c_0,\,c_1,\,\,c_2$ and the coefficients $a_n$, under which the differential equation $\dfrac{d^2w}{ds^2}+(b_0e^{s}+b_1)\dfrac{dw}{ds}+(c_0e^{2s}+c_1e^{s}+c_2)w=\sum\limits_{n=1}^{\infty}a_ne^{ns}$has an entire solution which is pseudostarlike or pseudoconvex of order $\alpha\in [0,\,1)$ and type $\beta \in (0,\,1]$. It is proved that by some conditions for such solution the asymptotic equality holds $\ln\,\max\{|F(\sigma+it)|\colon t\in {\mathbb R}\}=\dfrac{1+o(1)}{2}\left(|b_0|+\sqrt{|b_0|^2+4|c_0|}\right)$ as $\sigma \to+\infty$.

On close-to-pseudoconvex Dirichlet series

“…Using the Alexander criterion, S. Shah [4] indicated the conditions for real parameters γ 0 , γ 1 , γ 2 , under which a differential equation z 2 w ′′ +zw ′ +(γ 0 z 2 +γ 1 z +γ 2 )w = 0 has an entire transcendental solution f such that the function f and all its derivatives are close-to-convex in D. Many authors (see, for example, [5][6][7][8]) continued Shah's research.…”

On the solutions of some generalization of the Shah-type differential equation

“…under which there exists an entire transcendental solution (1) such that f and all its derivatives are close-to-convex in D. The convexity of solutions of the Shah equation has been studied in [9,10]. Substituting z = e s we obtain the dierential equation…”

Analytic vector-functions in the unit ball having bounded $\mathbf{L}$-index in joint variables

“…It is important because any function of bounded index have its growth estimates, local behavior of derivatives and some uniform distribution of zeros. Moreover, some authors [26][27][28][29][30][31][32][33] study connection between p-valence and l-index boundedness of analytic functions, the existence of solutions of the second order linear differential equations with polynomial coefficients which are starlike, convex, close-to-convex and of bounded lindex (l : C → R + is a continuous function). In other words, they combine analytic and geometric properties of functions of complex variable.…”