Successive derivatives of entire functions

Successive Derivatives of Entire Functions of Exponential Type

On the Number of Zeros of Successive Derivatives of Entire Functions of Finite Order

1 We use this occasion to point out that the condition lim inf log M (r) < 1 r-+c g(r) in Theorem 2 of [1] can be replaced by the somewhat weaker condition : there exists a, sequence r1, + oo such that log M (r,,) < g(r1,). It is clear from the proof that only this is actually used . Thus the following assertion is true THEOREM B. Let g(r) denote an arbitrary increasing function, defined in 0 <...

Proof of a conjecture of Pólya on the zeros of successive derivatives of real entire functions

We prove Pólya’s conjecture of 1943: For a real entire function of order greater than 2 with finitely many non-real zeros, the number of non-real zeros of the n-th derivative tends to infinity as n → ∞ . We use the saddle point method and potential theory, combined with the theory of analytic functions with positive imaginary part in the upper half-plane.

Entire Functions That Share Rational Functions with Their Derivatives

In this paper, we use the idea of normal family to deal with the uniqueness problems of entire functions that share a rational function with its derivative and get a uniqueness theorem. The conclusions in this paper can be used to improve several known results. Some examples are provided to show that the results presented in this paper are possible. AMS Mathematics Subject Classification (2010)...

Uniqueness of Entire Functions Sharing Polynomials with Their Derivatives

and Applied Analysis 3 From the ideas of Theorem D to Theorem F, it is natural to ask whether the values a, b in Theorem C can be replaced by two polynomials Q1, Q2. The main purpose of this paper is to investigate this problem. We prove the following result. Theorem 1.2. Let Q1 z a1z a1,p−1zp−1 · · · a1,0 and Q2 z a2z a2,p−1zp−1 · · · a2,0 be two polynomials such that degQ1 z degQ2 z p (where ...