On a generalization of some Shah equation - Sheremeta, M.M.; Trukhan, Yu.S. (Carpathian Mathematical Publications, 2024)
A Dirichlet series F(s)=e^hs+_k=2^∞f_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) and type β(0, 1] in Π_0={s: Re s<0} if \[|F'(s)F(s)-h|<β|F'(s)F(s)-(2α-h)|\] for all sΠ_0. Similarly, the function F is said to be pseudoconvex of order α[0, h) and type β(0, 1] if \[|F''(s)F'(s)-h|<β|F''(s)F'(s)-(2α-h)|\] for all sΠ_0, and F is said to be close-to-pseudoconvex if there exists a pseudoconvex (with α=0 and β=1) function Ψ such that Re{F'(s)/Ψ'(s)}>0 in Π_0. Conditions on parameters a_1, a_2, b_1, b_2, c_1, c_2, under which the differential equation \[d^n wds^n+(a_1 e^hs+a_2)dwds+(b_1e^hs+b_2) w=c_1e^hs+c_2, n2,\] has an entire solution pseudostarlike or pseudoconvex of order α[0, h) and type β(0, 1], or close-to-pseudoconvex in Π_0 are found. It is proved that for such solution \[M(σ,F)=(1+o(1))nn|b_1| he^hσ/nasσ →+∞,\] where M(σ,F)={|F(σ+it)|: tR}.