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Nonasymptotic results on distribution of zeros of the function E p (z,μ). (English) Zbl 0905.30004

The authors have obtained some nonasymptotic results on the distribution of zeros of the function

E ρ (z,μ)= k=0 z k Γ(μ+k/ρ),ρ>0,μ

which is a generalization of the classical Mittag-Leffler function. One of the main results of the article is contained in the following theorem.

Theorem. Assume that one of the following conditions is satisfied:

(i) ρ>1, μ[1,1+1/ρ]; (ii) ρ=1, μ(0,2); (iii) 1/2<ρ<1, μ[1/ρ-1,1][1/ρ,2].

Then all zeros of E ρ (z,μ) are situated outside the angle {z:|argz|π/2ρ}.

The set W={(ρ,μ): all zeros of E ρ (z,μ) are negative and simple} has been investigated in the article and as a corollary the generalization of Wiman’s theorem has been proved:

If 0<ρ1/2, then all zeros of both E ρ (z,1) and E ρ (z,2) are negative and simple.

30C15Zeros of polynomials, etc. (one complex variable)
30D15Special classes of entire functions; growth estimates
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