Abstract
In this paper, we investigate the admissible entire solutions of finite order of the differential-difference equations (f'(z))^{2}+P^{2}(z)f^{2}(z+c)=Q(z)e^{alpha(z)} and (f'(z))^{2}+[f(z+c)-f(z)]^{2}=Q(z)e^{alpha(z)}, where P(z), Q(z) are two non-zero polynomials, alpha(z) is a polynomial and cinmathbb{C}backslash{0}. In addition, we investigate the non-existence of entire solutions of finite order of the differential-difference equation (f'(z))^{n}+P(z)f^{m}(z+c)=Q(z), where P(z), Q(z) are two non-constant polynomials, cinmathbb{C}backslash{0}, m, n are positive integers and satisfy frac{1}{m}+frac{1}{n}<2 except for m=1, n=2.
Highlights
Introduction and main resultsIn this paper, we assume that the reader is familiar with the standard symbols and fundamental results of Nevanlinna theory [, ]
We investigate the admissible entire solutions of finite order of the differential-difference equations (f (z))2 + P2(z)f 2(z + c) = Q(z)eα(z) and (f (z))2 + [f (z + c) – f (z)]2 = Q(z)eα(z), where P(z), Q(z) are two non-zero polynomials, α(z) is a polynomial and c ∈ C\{0}
We denote by S(r, f ) any quantify satisfying S(r, f ) = o(T(r, f )), as r → ∞, outside of a possible exceptional set of finite logarithmic measure
Summary
We assume that the reader is familiar with the standard symbols and fundamental results of Nevanlinna theory [ , ]. If the differential-difference equation f (z) + P (z)f (z + c) = Q(z) admits a transcendental entire solution of finite order, P(z), Q(z) reduce to constants, and peaz+b – qe–(az+b). If the differential-difference equation f (z) + P (z)f (z + c) = Q(z)eα(z) admits a transcendental entire solution of finite order, f (z) must satisfy one of the following cases:. Let P(z), Q(z) be two non-constant polynomials and c ∈ C\{ }, the equation f (z) n + P(z)f m(z + c) = Q(z) has no transcendental entire solutions with finite order, provided that m = , n = , where m, n are positive integers.
Sign-in/Register to view highlights & summary 
Published Version (
Free)
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call