Singular Direction and q -Difference Operator of Meromorphic Functions
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Singular Direction and q-Difference Operator of Meromorphic Functions Jianren Long1,2 · Jianyong Qiao2 · Xiao Yao3 Received: 26 June 2019 / Revised: 2 December 2019 © Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2020
Abstract We study the common singular direction problem of meromorphic function for qdifference version operator; some criterions of the existence of common singular direction have been established. Further, the common singular direction of solutions of q-difference equations is also discussed in this paper. Keywords Borel direction · Julia direction · Nevanlinna theory · q-difference operator · q-difference equation Mathematics Subject Classification Primary 30D35; Secondary 30D30 · 39A13
1 Introduction We use the standard notations of Nevanlinna theory in this paper, such as the Nevanlinna characteristic T (r , f ); we refer [18,31] for the details. Let f be a meromorphic function, and we define its order of growth ρ( f ) and lower order of growth μ( f ) by
Communicated by V. Ravichandran.
B
Jianren Long [email protected]; [email protected] Jianyong Qiao [email protected] Xiao Yao [email protected]
1
School of Mathematical Sciences, Guizhou Normal University, Guiyang 550001, People’s Republic of China
2
School of Computer Sciences and School of Sciences, Beijing University of Posts and Telecommunications, Beijing 100876, People’s Republic of China
3
School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, People’s Republic of China
123
J. Long et al.
ρ( f ) := lim sup r →∞
log T (r , f ) , log r
μ( f ) := lim inf r →∞
log T (r , f ) , log r
respectively. If f is an entire function, then T (r , f ) can be replaced with log M(r , f ) above, where M(r , f ) = max|z|=r | f (z)|. Given θ ∈ [0, 2π ), for any > 0, setting (θ −, θ +) := {z : | arg z −θ | < } and n(r , (θ −, θ +), a, f ) be the number of a-valued point of f in (θ − , θ + ) {z : |z| < r } counting multiplicities for ˆ any a ∈ C. The ray arg z = θ is called to be a Julia direction of f if lim sup n(r , (θ − , θ + ), a, f ) = +∞ r →∞
ˆ with at most two exceptions. If f is of finite positive order ρ, then we for all a ∈ C can define its Borel direction as follows. The ray arg z = θ is called to be a Borel direction of order ρ of f if lim sup r →∞
log n(r , (θ − , θ + ), a, f ) =1 ρ log r
ˆ with at most two exceptions. In the following, for short, we call the Borel for all a ∈ C direction of order ρ as the Borel direction. It is well known that the Borel direction is always a Julia direction, while the converse is not true. We refer [1,13,31,34] for the topics of Julia direction and Borel direction. Indeed, for the case of infinite-order meromorphic functions, in order to give a better generalization of Borel direction into this case, one need to modify the definition of Borel direction above with ρ(r ) log r to replace the ρ log r above, where ρ(r ) is its proximate order which the definition is given in Sect. 2. In 1928, G. Valiron [28] posed
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