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

Анотація

A Dirichlet series F(s)=e^hs+∑^∞_k=2f_ke^sλ_k with the exponents 0<h<λ_k↑+∞ and the abscissa of absolute convergence  σ_a[F]≥0 is said to be pseudostarlike of order α∈[0,h) in Π₀={s: Re s<0} if Re{F'(s)/F(s)}>α for all s∈Π₀. Similarly, the function F is said to be pseudoconvex of order α∈[0,h) if Re{F''(s)/F'(s)}>α for all s∈Π₀.
The  equation dⁿw/dsⁿ+(∑ⁿ_j=0γ_j e^hjs)w=ae^hs is considered, where n≥3, h>0 and a≠=0. It is proved that if hⁿ+γ₀>0$ then this equation has a solution
F(s)=(a⋅e^hs)/(hⁿ+γ₀)+∑^∞_k=2f_ke^hsk,
where f_k=-1/((hk)ⁿ+γ₀)∑_j=1^min{k-1,n} γ_jf_k-j for k≥2 (Lemma 1). For n=3, a=h³+γ₀ conditions on parameters γ_j, under which the function F is pseudostarlike (Theorem 1) or pseudoconvex of order α∈[0,h) (Theorem 2) are found.