Complex oscillation of meromorphic solutions for difference Riccati equation

Advances in Difference Equations, Sep 2014

In this paper, we investigate zeros and α-points of meromorphic solutions f ( z ) for difference Riccati equations, and we obtain some estimates of exponents of convergence of zeros and α-points of f ( z ) and shifts f ( z + n ) , differences Δ f ( z ) = f ( z + 1 ) − f ( z ) , and divided differences Δ f ( z ) f ( z ) . MSC: 30D35, 39B12.

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Complex oscillation of meromorphic solutions for difference Riccati equation

Yang-Yang Jiang 0 Zhi-Qiang Mao 0 Min Wen 1 0 School of Mathematics and Computer, Jiangxi Science and Technology Normal University , Nangchang, Jiangxi , China 1 Department of Civil and Architectural Engineering, Nanchang Institute of Technology , Nanchang, Jiangxi 330099 , China In this paper, we investigate zeros and -points of meromorphic solutions f (z) for difference Riccati equations, and we obtain some estimates of exponents of convergence of zeros and -points of f (z) and shifts f (z + n), differences f (z) = f (z + 1) - f (z), and divided differences f (fz()z) . MSC: 30D35; 39B12 1 Introduction and main results In this paper, we assume that the reader is familiar with the standard notations and basic results of Nevanlinna's value distribution theory (see [, ]). In addition, we use the notions (f ) to denote the order of growth of the meromorphic function f (z), (f ), and ( f ) to denote the exponents of convergence of zeros and poles of f (z), respectively. We say a meromorphic function f (z) is oscillatory if f (z) has infinitely many zeros. The theory of difference equations, the methods used in their solutions, and their wide applications have advanced beyond their adolescent stage to occupy a central position in applicable analysis. The theory of oscillation play an important role in the research on discrete equations, and it is systematically introduced in []. The complex oscillation is the development and deepening of the corresponding real oscillation, and it can profoundly reveals the essence of the oscillation problem that the property of oscillation is investigated in complex domain. Recently, as the difference analogs of Nevanlinna's theory were being investigated [-], many results on the complex difference equations have been got rapidly. Many papers [, -] mainly deal with the growth of meromorphic solutions of some difference equations, and several papers [, , -] deal with analytic properties of meromorphic solutions of some nonlinear difference equations. Especially, there has been an increasing interest in studying difference Riccati equations in the complex plane [, , , ]. In [], Ishizaki gave some surveys of the basic properties of the difference Riccati equation - y(z + ) = where A(z) is a rational function, which have analogs in the differential case []. In the proof of the celebrated classification theorem, Halburd and Korhonen [] were concerned with the difference Riccati equation of the form where A is a polynomial, = . In [], Chen and Shon investigated the existence and forms of rational solutions, and the Borel exceptional value, zeros, poles, and fixed points of transcendental solutions, and they proved the following theorem. Theorem A Let = be a constant and A(z) = mn((zz)) be an irreducible nonconstant rational function, where m(z) and n(z) are polynomials with deg m(z) = m and deg n(z) = n. If f (z) is a transcendental finite order meromorphic solution of the difference Riccati equation then (i) if (f ) > , then f (z) has at most one Borel exceptional value; (ii) ( f ) = (f ) = (f ); (iii) if A(z) z z + , then the exponent of convergence of fixed points of f (z) satisfies (f ) = (f ). In [], the first author investigated fixed points of meromorphic functions f (z) for difference Riccati equation (), and obtain some estimates of exponents of convergence of fixed points of f (z) and shifts f (z + n), differences f (z) = f (z + ) f (z), and divided differences f f(z(z)) . In this paper, we investigate zeros and -points of meromorphic solutions f (z) for difference Riccati equations (), and we obtain some estimates of the exponents of convergence of zeros and -points of f (z) and shifts f (z + n), differences f (z) = f (z + ) f (z), and divided differences f f(z(z)) of meromorphic solutions of (). We prove the following theorem. Theorem . Let = be a constant and A(z) be a nonconstant rational function. Set f (z) = f (z + ) f (z). If there exists a nonconstant rational function s(z) such that A(z) = s(z), then every finite order transcendental meromorphic solution f (z) of the difference Riccati equation (), its difference f (z), and divided difference f f(z(z)) are oscillatory and satisfy Theorem . Let A(z) be a nonconstant rational function. If is a non-zero complex constant, then every finite order transcendental meromorphic solution f (z) of the difference Riccati equation f (z + ) = A(z) +f(fz()z) (i) if = , then (f (z + n) ) = (f ), n = , , , . . . ; (ii) if there is a rational function n(z) satisfying A(z) = ( + ) ( + )n(z), then ( f f(z(z)) ) = (f ); (iii) if there is a rational function m(z) satisfying f (z + ) = A(z) +f(fz()z) , where A(z) = z(z+) , Q(z) is a periodic function with period . Note that for any [, +), there exists a prime periodic entire function Q(z) of order (Q) = by Ozawa []. Thus (f ) = (Q) = . Also, this solution f (z) = zQQ((zz))+zz((zz))((zz++)) satisfies f f (z(z)) = (z + )[Qz(z()z+z)(z[Q)((zz)+)]z[(Q (...truncated)


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Yang-Yang Jiang, Zhi-Qiang Mao, Min Wen. Complex oscillation of meromorphic solutions for difference Riccati equation, Advances in Difference Equations, 2014, pp. 247, 2014,