Meromorphic functions of finite \varphi-order and linear Askey-Wilson divided difference equations
aa r X i v : . [ m a t h . C V ] J a n MEROMORPHIC FUNCTIONS OF FINITE ϕ -ORDER ANDLINEAR ASKEY-WILSON DIVIDED DIFFERENCEEQUATIONS HUI YU, JANNE HEITTOKANGAS ∗ , JUN WANG, AND ZHI-TAO WEN Abstract.
The growth of meromorphic solutions of linear difference equa-tions containing Askey-Wilson divided difference operators is estimated.The ϕ -order is used as a general growth indicator, which covers the growthspectrum between the logarithmic order ρ log ( f ) and the classical order ρ ( f ) of a meromorphic function f . Key words:
Askey-Wilson divided difference operator, Askey-Wilsondivided difference equation, lemma on the logarithmic difference, mero-morphic function, ϕ -order. MSC 2020:
Primary 39A13; Secondary 30D35. Introduction
Suppose that q is a complex number satisfying 0 < | q | <
1. In 1985, Askeyand Wilson evaluated a q -beta integral [1, Theorem 2.1], which allowed themto construct a family of orthogonal polynomials [1, Theorems 2.2–2.5]. Thesepolynomials are eigensolutions of a second order difference equation [1, p. 36]that involves a divided difference operator D q currently known as the Askey-Wilson operator . We will define D q below and call it the AW-operator forbrevity. In general, any three consecutive orthogonal polynomials satisfy acertain three term recurrence relation, see [1, p. 4] or [8, p. 42].Recently, Chiang and Feng [3] have obtained a full-fledged Nevanlinna the-ory for meromorphic functions of finite logarithmic order with respect to theAW-operator on the complex plane C . The concluding remarks in [3] admitthat the logarithmic order of growth appears to be restrictive, even thoughthis class contains a large family of important meromorphic functions. Thisencourages us to generalize some of the results in [3] in such a way that theassociated results for finite logarithmic order follow as special cases.Let ϕ : ( R , ∞ ) → (0 , ∞ ) be a non-decreasing unbounded function. The ϕ -order of a meromorphic function f in C was introduced in [5] as the quantity ρ ϕ ( f ) = lim sup r →∞ log T ( r, f )log ϕ ( r ) . Prior to [5], the ϕ -order was used as a growth indicator for meromorphicfunctions in the unit disc in [4]. In the plane case, the logarithmic order ρ log ( f ) and the classical order ρ ( f ) of f follow as special cases when choosing ∗ Corresponding author. H. Yu, J. Heittokangas, J. Wang, and Z. T. Wen ϕ ( r ) = log r and ϕ ( r ) = r , respectively. This leads us to impose a globalgrowth restriction log r ≤ ϕ ( r ) ≤ r, r ≥ R . (1.1)Here and from now on, the notation r ≥ R is being used to express that theassociated inequality is valid ”for all r large enough”.For an entire function f , the Nevanlinna characteristic T ( r, f ) can be re-placed with the logarithmic maximum modulus log M ( r, f ) in the quanti-ties ρ ( f ) and ρ log ( f ) by using a well-known relation between T ( r, f ) andlog M ( r, f ), see [7, p. 23]. The same is true for the ϕ -order, namely ρ ϕ ( f ) = lim sup r →∞ log log M ( r, f )log ϕ ( r ) , (1.2)provided that ϕ is subadditive, that is, ϕ ( a + b ) ≤ ϕ ( a )+ ϕ ( b ) for all a, b ≥ R .In particular, this gives ϕ (2 r ) ≤ ϕ ( r ), which yields (1.2). Moreover, up to anormalization, subadditivity is implied by concavity, see [5] for details.Following the notation in [1] (see [3] and [6, p. 300] for an alternativenotation), we suppose that f ( x ) is a meromorphic function in C , and let x = cos θ and z = e iθ , where θ ∈ C . Then, for x = ±
1, the AW-operator isdefined by( D q f )( x ) := ˘ f ( q e iθ ) − ˘ f ( q − e iθ )˘ e ( q e iθ ) − ˘ e ( q − e iθ ) = ˘ f ( q e iθ ) − ˘ f ( q − e iθ )( q − q − )( z − /z ) / , (1.3)where x = ( z + 1 /z ) / θ , z = e iθ , e ( x ) = x and˘ f ( z ) = f (( z + 1 /z ) /
2) = f ( x ) = f (cos θ ) . In the exceptional cases x = ±
1, we define( D q f )( ±
1) = lim x →± x = ± ( D q f )( x ) = f ′ ( ± ( q + q − ) / . The branch of the square root in z = x + √ x − x ∈ C there corresponds a unique z ∈ C , see [3] and [6, p. 300].It is known that D q f is meromorphic for a meromorphic function f and entirefor an entire function f [3, Theorem 2.1]. The AW-operator in (1.3) can bewritten in the alternative form( D q f )( x ) = f (ˆ x ) − f (ˇ x )ˆ x − ˇ x , where x = ( z + 1 /z ) / θ andˆ x = q z + q − z − , ˇ x = q − z + q z − . Finally, AW-operators of arbitrary order are defined by D q f = f and D nq f = D q ( D n − q f ), where n ∈ N .Lemma A below is a pointwise AW-type lemma on the logarithmic dif-ference proved in [3, Lemma 4.2], and it is used in [3] to study the growthof meromorphic solutions of Askey-Wilson divided difference equations. Wenote that finite logarithmic order implies finite ϕ -order because of the growthrestriction (1.1). he ϕ -order for the Askey-Wilson divided differences Lemma A.
Let f ( x ) be a meromorphic function of finite logarithmic ordersuch that D q f , and let α ∈ (0 , be arbitrary. Then there exists aconstant C α > such that for | q / | + | q − / | ) | x | < R , we have log + (cid:12)(cid:12)(cid:12)(cid:12) D q f ( x ) f ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ R ( | q / − | + | q − / − | ) | x | ( R − | x | )( R − | q / | + | q − / | ) | x | ) ( m ( R, f ) + m ( R, /f ))+ 2( | q / − | + | q − / − | ) | x | (cid:18) R − | x | + 1 R − | q / | + | q − / | ) | x | (cid:19) × ( n ( R, f ) + n ( R, /f ))+ 2 C α ( | q / − | α + | q − / − | α ) | x | α X | c n |
0. The estimate (1.5) in turn is used to prove agrowth estimate [3, Theorem 12.4] for meromorphic solutions of AW-divideddifference equations, stated as follows.
Theorem B.
Let a ( x ) , a ( x ) , . . . , a n − ( x ) be entire functions such that ρ log ( a ) > max ≤ j ≤ n { ρ log ( a j ) } . Suppose that f is an entire solution of the AW-divided difference equation n X j =0 a j ( x ) D jq f ( x ) = 0 , where a n ( x ) = 1 . Then ρ log ( f ) ≥ ρ log ( a ) + 1 . Our main objectives are to find ϕ -order analogues of the estimate (1.5) andof Theorem B. A non-decreasing function s : ( R , ∞ ) → (0 , ∞ ) satisfying aglobal growth restriction r < s ( r ) ≤ r , r ≥ R , (1.6)will take the role of R in Lemma A. Suitable test functions for ϕ and s thenare, for example, ϕ ( r ) = log α r, ϕ ( r ) = exp(log β r ) , ϕ ( r ) = r β , H. Yu, J. Heittokangas, J. Wang, and Z. T. Wen along with s ( r ) = r log r and s ( r ) = r α , where α ∈ (1 ,
2] and β ∈ (0 , ϕ -order is given in Section 2. TwoAW-type lemmas on the logarithmic difference in terms of the ϕ -order aregiven in Section 3. One of them will be among the most important individualtools later on. Section 4 consists of lemmas on AW-type counting functions aswell as on the Nevanlinna characteristic of D q f . These lemmas are crucial inproving the main results, which are Theorem 2.1 and 2.2 below. The detailsof the proofs are given in Section 5.2. Results on Askey-Wilson divided difference equations
We consider the growth of meromorphic solutions of AW-divided differenceequations n X j =0 a j ( x ) D jq f ( x ) = 0 (2.1)and of the corresponding non-homogeneous AW-divided difference equations n X j =0 a j ( x ) D jq f ( x ) = a n +1 ( x ) , (2.2)where a , . . . , a n +1 are meromorphic functions, and a a n
0. The resultsthat follow depend on growth parameters introduced in [5] and defined by α ϕ,s = lim inf r →∞ log ϕ ( r )log ϕ ( s ( r )) and γ ϕ,s = lim inf r →∞ log log s ( r ) r log ϕ ( r ) . (2.3)Due to the assumptions (1.1) and (1.6), we always have α ϕ,s ∈ [0 ,
1] and γ ϕ,s ∈ [ −∞ , r →∞ s ( r ) r > , which ensures that γ ϕ,s ∈ [0 , α ϕ,s and γ ϕ,s can be found in [5].Theorem 2.1 below reduces to Theorem B when choosing ϕ ( r ) = log r and s ( r ) = r and when the coefficients and solutions are entire functions. Theorem 2.1.
Suppose that ϕ ( r ) is subadditive, and let α ϕ,s and γ ϕ,s be theconstants in (2.3) . Let a , . . . , a n be meromorphic functions of finite ϕ -ordersuch that ρ ϕ ( a ) > max ≤ j ≤ n { ρ ϕ ( a j ) } . (a) Suppose that lim sup r →∞ s ( r ) r = ∞ and that s ( r ) is convex and differen-tiable. If f is a non-constant meromorphic solution of (2.1) , then ρ ϕ ( f ) ≥ α nϕ,s ρ ϕ ( a ) . (2.4) Moreover, if the coefficients a , . . . , a n are entire, then ρ ϕ ( f ) ≥ α nϕ,s ρ ϕ ( a ) + α nϕ,s γ ϕ,s . (2.5) he ϕ -order for the Askey-Wilson divided differences Suppose that lim sup r →∞ s ( r ) r < ∞ . If f is a non-constant meromorphicsolution of (2.1) , then ρ ϕ ( f ) ≥ α n − ϕ,s ρ ϕ ( a ) . Remark 1.
For certain ϕ ( r ), for example, for ϕ ( r ) = log α r , where α ∈ (1 , s ( r ). If the coefficients a , . . . , a n are entire, then itfollows from (2.3) and (2.5) that ρ ϕ ( f ) ≥ ρ ϕ ( a )+1 /α in Theorem 2.1(a) whenchoosing s ( r ) = r , which is stronger than the conclusion ρ ϕ ( f ) ≥ ρ ϕ ( a ) inTheorem 2.1(b) when choosing s ( r ) = 2 r .On the other hand, the opposite is true for some suitable ϕ ( r ). For instance,choose ϕ ( r ) = r β , where β ∈ (0 , s ( r ) = 2 r and s ( r ) = r ,respectively. Then we get ρ ϕ ( f ) ≥ ρ ϕ ( a ) from Theorem 2.1(b), which isstronger than the conclusion ρ ϕ ( f ) ≥ (1 / n ρ ϕ ( a ) in Theorem 2.1(a), whichin turn follows from (2.3) and (2.4).The following result is a growth estimate for meromorphic solutions of thenon-homogeneous equations (2.2). Theorem 2.2.
Suppose that ϕ ( r ) is subadditive. Let a , . . . , a n be meromor-phic functions of finite ϕ -order such that ρ ϕ ( a ) > max ≤ j ≤ n +1 { ρ ϕ ( a j ) } . If f is a non-constant meromorphic solution of (2.2) , then ρ ϕ ( f ) ≥ α n − ϕ,s ρ ϕ ( a ) . The proofs of Theorems 2.1 and 2.2 in Section 5 are based on an AW-type lemma on the logarithmic difference discussed in Section 3 as well as onestimates for AW-type counting functions discussed in Section 4.3.
Estimates for the Askey-Wilson typelogarithmic difference
Lemma 3.1 below is an AW-type lemma on the logarithmic difference, whichreduces to [3, Theorem 3.1] when choosing ϕ ( r ) = log r and s ( r ) = r . Theproof uses the notation g ( r ) . h ( r ) to express that there exists a constant C ≥ g ( r ) ≤ Ch ( r ) for all r ≥ R . Lemma 3.1.
Let f be a meromorphic function of finite ϕ -order ρ ϕ ( f ) suchthat D q f . Let α ϕ,s and γ ϕ,s be the constants in (2.3) , let ε > , anddenote | x | = r . (a) If lim sup r →∞ s ( r ) r = ∞ and if s ( r ) is convex and differentiable, then m (cid:18) r, D q f ( x ) f ( x ) (cid:19) = O ϕ ( s ( r )) ρ ϕ ( f )+ ε log s ( r ) r + 1 ! = O (cid:0) ϕ ( s ( r )) ρ ϕ ( f ) − α ϕ,s γ ϕ,s + ε (cid:1) . (b) If lim sup r →∞ s ( r ) r < ∞ and if ϕ ( r ) is subadditive, then m (cid:18) r, D q f ( x ) f ( x ) (cid:19) = O (cid:0) ϕ ( r ) ρ ϕ ( f )+ ε (cid:1) . H. Yu, J. Heittokangas, J. Wang, and Z. T. Wen
Proof. (a) By the proof of [5, Lemma 3.1(a)], there exist non-decreasing func-tions u, v : [1 , ∞ ) → (0 , ∞ ) with the following properties:(1) r < u ( r ) < s ( r ) and r < v ( r ) < s ( r ) for all r ≥ R ,(2) u ( r ) /r → ∞ and v ( r ) /r → ∞ as r → ∞ ,(3) 2 − s ( r ) ≤ v ( u ( r )) ≤ s ( r ) for all r ≥ R ,(4) 2 log( u ( r ) /r ) ≤ log( s ( r ) /r ) ≤ u ( r ) /r for all r ≥ R .Using the standard estimate N ( v ( r ) , f ) − N ( r, f ) = Z v ( r ) r n ( t, f ) t dt ≥ n ( r, f ) log v ( r ) r and the properties (3) and (4), we deduce that n ( u ( r ) , f ) ≤ T ( s ( r ) , f )log s ( r )2 r − log u ( r ) r . T ( s ( r ) , f )log s ( r ) r . ϕ ( s ( r )) ρ ϕ ( f )+ ε log s ( r ) r , (3.1)and similarly for n ( u ( r ) , /f ). Choose R = u ( r ). We integrate (1.4) from 0to 2 π , and we make use of the properties (1) and (4) together with (3.1) andformulas (63)–(64) in [3], and obtain m (cid:18) r, D q f ( x ) f ( x ) (cid:19) . T ( u ( r ) , f ) u ( r ) /r + n ( u ( r ) , f ) + n ( u ( r ) , /f ) + 1 . ϕ ( s ( r )) ρ ϕ ( f )+ ε log s ( r ) r + 1 . This proves the first identity in Case (a).From (2.3), we get α ϕ,s γ ϕ,s ≤ lim inf r →∞ log ϕ ( r )log ϕ ( s ( r )) · log log s ( r ) r log ϕ ( r ) ! = lim inf r →∞ log log s ( r ) r log ϕ ( s ( r )) , and so log s ( r ) r ≥ ϕ ( s ( r )) α ϕ,s γ ϕ,s − ε , r ≥ R . Recall from [5, Corollary 4.3] that, for a non-constant meromorphic function f of finite ϕ -order ρ ϕ ( f ), we have ρ ϕ ( f ) ≥ α ϕ,s γ ϕ,s . Thus ϕ ( s ( r )) ρ ϕ ( f )+ ε log s ( r ) r ≤ ϕ ( s ( r )) ρ ϕ ( f ) − α ϕ,s γ ϕ,s + ε , r ≥ R , (3.2)where the right-hand side tends to infinity as r → ∞ . This proves the secondidentity in Case (a).(b) By the assumptions on s ( r ), there exists a constant C ∈ (1 , ∞ ) suchthat r < s ( r ) < Cr for all r ≥ R . We choose R = Br , where B = max { [ C ] , [2( | q / | + | q − / | )] } + 1 (3.3)is an integer. Integrating (1.4) from 0 to 2 π and making use of formulas(63)–(64) in [3] together with T (2 r, f ) ≥ Z rr n ( t, f ) t dt ≥ n ( r, f ) log 2 , (3.4) he ϕ -order for the Askey-Wilson divided differences m (cid:18) r, D q f ( x ) f ( x ) (cid:19) . T ( Br, f ) + n ( Br, f ) + n ( Br, /f ) + 1 . ϕ (2 Br ) ρ ϕ ( f )+ ε + 1 . (3.5)Since the subadditivity of ϕ yields ϕ (2 Br ) ≤ Bϕ ( r ), the assertion followsfrom (3.5). This completes the proof. (cid:3) Lemma 3.2 below is a pointwise estimate for the AW-type logarithmicdifference that holds outside of an exceptional set. The result reduces to[3, Theorem 3.2] when choosing ϕ ( r ) = log r and s ( r ) = r . Lemma 3.2.
Let f be a meromorphic function of finite ϕ -order ρ ϕ ( f ) suchthat D q f . Let α ϕ,s > and γ ϕ,s be the constants in (2.3) , let ε > , anddenote | x | = r . Suppose that ϕ ( r ) is continuous and satisfies lim sup r →∞ log ϕ ( r )log r = 0 . (3.6)(a) If lim sup r →∞ s ( r ) r = ∞ and if s ( r ) is convex and differentiable, then log + (cid:12)(cid:12)(cid:12)(cid:12) D q f ( x ) f ( x ) (cid:12)(cid:12)(cid:12)(cid:12) = O ϕ ( s ( r )) ρ ϕ ( f )+ ε log s ( r ) r + 1 ! = O (cid:0) ϕ ( s ( r )) ρ ϕ ( f ) − α ϕ,s γ ϕ,s + ε (cid:1) holds outside of an exceptional set of finite logarithmic measure. (b) If lim sup r →∞ s ( r ) r < ∞ and if ϕ ( r ) is subadditive, then log + (cid:12)(cid:12)(cid:12)(cid:12) D q f ( x ) f ( x ) (cid:12)(cid:12)(cid:12)(cid:12) = O (cid:0) ϕ ( r ) ρ ϕ ( f )+ ε (cid:1) holds outside of an exceptional set of finite logarithmic measure.Proof. We modify the proof of [3, Theorem 3.2] as follows.(a) Denote { d n } := { c n } ∪ { q / c n } ∪ { q − / c n } , (3.7)where { c n } is the combined sequence of zeros and poles of f . Let E n = r : r ∈ | d n | − | d n | ϕ ( | d n | + 3) ρϕ ( f )+ εαϕ,s , | d n | + | d n | ϕ ( | d n | + 3) ρϕ ( f )+ εαϕ,s and E = ∪ n E n , where α ϕ,s ∈ (0 ,
1] is defined in (2.3). In what follows, weconsider r E . We proceed to prove that | x − d n | ≥ | x | ϕ ( | x | + 3) ρϕ ( f )+ εαϕ,s , | x | = r ≥ R . (3.8)The proof is divided into three cases in each of which | x | ≥ R . H. Yu, J. Heittokangas, J. Wang, and Z. T. Wen (1) Suppose that | x | < | d n | − | d n | ϕ ( | d n | +3) ρϕ ( f )+ εαϕ,s . From (3.6), the function | x | ϕ ( | x | +3) ρϕ ( f )+ εαϕ,s is increasing, and so | x − d n | ≥ || x | − | d n || ≥ | d n | ϕ ( | d n | + 3) ρϕ ( f )+ εαϕ,s ≥ | x | ϕ ( | x | + 3) ρϕ ( f )+ εαϕ,s . (2) Suppose that | d n | + | d n | ϕ ( | d n | +3) ρϕ ( f )+ εαϕ,s ≤ | x | − | x | ϕ ( | x | +3) ρϕ ( f )+ εαϕ,s . Clearly, | x − d n | ≥ | d n | ϕ ( | d n | + 3) ρϕ ( f )+ εαϕ,s + | x | ϕ ( | x | + 3) ρϕ ( f )+ εαϕ,s ≥ | x | ϕ ( | x | + 3) ρϕ ( f )+ εαϕ,s . (3) Suppose that | d n | + | d n | ϕ ( | d n | +3) ρϕ ( f )+ εαϕ,s < | x | and | x | − | x | ϕ ( | x | + 3) ρϕ ( f )+ εαϕ,s ≤ | d n | + | d n | ϕ ( | d n | + 3) ρϕ ( f )+ εαϕ,s . Then we have | x − d n | ≥ | d n | ϕ ( | d n | +3) ρϕ ( f )+ εαϕ,s and | x | = | d n | (1 + o (1)) as | x | → ∞ (or as n → ∞ ). This yields (3.8) by the continuity of ϕ ( r ).Keeping in mind that r E , this completes the proof of (3.8).Let α ∈ (0 , X | c n |
0, we have ϕ ( | d N | ) ρϕ ( f )+ εαϕ,s < δ . Using the fact that log(1 + | x | ) ≤ | x | for all | x | ≥
0, the constant C δ = − δ > ϕ ( | d N | ) ρϕ ( f )+ εαϕ,s − ϕ ( | d N | ) ρϕ ( f )+ εαϕ,s ≤ C δ · ϕ ( | d N | ) ρϕ ( f )+ εαϕ,s , N ≥ R . Therefore,log-meas ( E ) = (cid:18)Z E ∩ [1 , | d N | ] + Z E ∩ [ | d N | , ∞ ) (cid:19) dtt ≤ log | d N | + ∞ X n = N Z E n dtt = log | d N | + ∞ X n = N log 1 + ϕ ( | d n | ) ρϕ ( f )+ εαϕ,s − ϕ ( | d n | ) ρϕ ( f )+ εαϕ,s ≤ log | d N | + C δ ∞ X n = N ϕ ( | d n | ) λ ϕ + εαϕ,s < ∞ , which yields the assertion.(b) By making use of the proof of Lemma 3.1(b) and following the samemethod as in Case (a) above, we obtain (3.9) and (3.10). Choose R = Br and α = α ϕ,s ε ρ ϕ ( f )+ ε ) ∈ (0 , B is defined in (3.3). Then by substituting0 H. Yu, J. Heittokangas, J. Wang, and Z. T. Wen (3.4), (3.9) and (3.10) into (1.4), we havelog + (cid:12)(cid:12)(cid:12)(cid:12) D q f ( x ) f ( x ) (cid:12)(cid:12)(cid:12)(cid:12) . T ( Br, f ) + n ( Br, f ) + n ( Br, /f )+ ϕ ( r + 3) α αϕ,s ( ρ ϕ ( f )+ ε ) · ϕ (2 Br ) ρ ϕ ( f )+ ε + 1 ≤ ϕ (2 Br ) ρ ϕ ( f )+ ε , r E. Then the assertion follows from the subadditivity of ϕ , that is, ϕ (2 Br ) ≤ Bϕ ( r ). Similarly as in Case (a) above, we deduce that the logarithmicmeasure of the exceptional set E is finite. This completes the proof. (cid:3) Askey-Wilson type counting functionsand characteristic functions
In this section we state three lemmas, whose proofs are just minor modifi-cations of the corresponding results in [3]. For a non-constant meromorphicfunction f , it follows from [5, Lemmas 4.1–4.2] that ρ ϕ ( f ) ≥ α ϕ,s λ ϕ + α ϕ,s γ ϕ,s and, if α ϕ,s >
0, then n ( r, a, f ) = O ( ϕ ( r ) λ ϕ + ε ) ≤ O (cid:18) ϕ ( r ) ρϕ ( f ) αϕ,s − γ ϕ,s + ε (cid:19) , where λ ϕ is the ϕ -exponent of convergence of the a -points of f .Lemma 4.1 below is essential in proving Lemma 4.2, and it reduces to [3,Theorem 5.1] when choosing ϕ ( r ) = log r and s ( r ) = r . Lemma 4.1.
Let f be a non-constant meromorphic function of finite ϕ -order ρ ϕ ( f ) . Suppose that ϕ ( r ) is subadditive. Let α ϕ,s > and γ ϕ,s be the constantsin (2.3) , and let ε > and a ∈ b C . (a) If lim sup r →∞ s ( r ) r = ∞ and if s ( r ) is convex and differentiable, then N ( r, a, f (ˆ x )) = N ( r, a, f ( x )) + O (cid:18) ϕ ( r ) ρϕ ( f ) αϕ,s − γ ϕ,s + ε (cid:19) + O (log r ) ,N ( r, a, f (ˇ x )) = N ( r, a, f ( x )) + O (cid:18) ϕ ( r ) ρϕ ( f ) αϕ,s − γ ϕ,s + ε (cid:19) + O (log r ) . (b) If lim sup r →∞ s ( r ) r < ∞ , then N ( r, a, f (ˆ x )) = N ( r, a, f ( x )) + O (cid:0) ϕ ( r ) ρ ϕ ( f )+ ε (cid:1) + O (log r ) ,N ( r, a, f (ˇ x )) = N ( r, a, f ( x )) + O (cid:0) ϕ ( r ) ρ ϕ ( f )+ ε (cid:1) + O (log r ) . Lemma 4.2 below is a direct consequence of Lemma 4.1 and the definitionof the AW-operator D q f , and it reduces to [3, Theorem 3.3] when choosing ϕ ( r ) = log r and s ( r ) = r . Lemma 4.2.
Let f be a non-constant meromorphic function of finite ϕ -order ρ ϕ ( f ) . Suppose that ϕ ( r ) is subadditive. Let α ϕ,s > and γ ϕ,s be the constantsin (2.3) , and let ε > . he ϕ -order for the Askey-Wilson divided differences If lim sup r →∞ s ( r ) r = ∞ and if s ( r ) is convex and differentiable, then N ( r, D q f ) ≤ N ( r, f ) + O (cid:18) ϕ ( r ) ρϕ ( f ) αϕ,s − γ ϕ,s + ε (cid:19) + O (log r ) . (b) If lim sup r →∞ s ( r ) r < ∞ , then N ( r, D q f ) ≤ N ( r, f ) + O (cid:0) ϕ ( r ) ρ ϕ ( f )+ ε (cid:1) + O (log r ) . The following result reduces to [3, Theorem 3.4] when choosing ϕ ( r ) = log r and s ( r ) = r . Lemma 4.3.
Let f be a non-constant meromorphic function of finite ϕ -order ρ ϕ ( f ) . Suppose that ϕ ( r ) is subadditive. Let α ϕ,s > and γ ϕ,s be the constantsin (2.3) , and let ε ∈ (0 , . (a) If lim sup r →∞ s ( r ) r = ∞ and if s ( r ) is convex and differentiable, then T ( r, D q f ) ≤ T ( r, f ) + O (cid:18) ϕ ( r ) ρϕ ( f ) αϕ,s − γ ϕ,s + ε (cid:19) + O (log r ) . (b) If lim sup r →∞ s ( r ) r < ∞ , then T ( r, D q f ) ≤ T ( r, f ) + O (cid:0) ϕ ( r ) ρ ϕ ( f )+ ε (cid:1) + O (log r ) . Proof.
Choose ε ∗ = α ϕ,s ε ρ ϕ ( f )+ α ϕ,s ε ) ∈ (cid:0) , α ϕ,s (cid:1) . By the definition of the constant α ϕ,s in (2.3), it follows that ϕ ( s ( r )) ≤ ϕ ( r ) αϕ,s − ε ∗ , r ≥ R . (4.1)We replace ε in Lemma 3.1(a) with ε ′ = α ϕ,s ε + γ ϕ,s ε ∗ = (cid:16) α ϕ,s + α ϕ,s γ ϕ,s ε ρ ϕ ( f )+ α ϕ,s ε ) (cid:17) ε ,which we are allowed to do since 0 < α ϕ,s ≤ α ϕ,s + α ϕ,s γ ϕ,s ε ρ ϕ ( f )+ α ϕ,s ε ) < α ϕ,s ≤ ϕ ( s ( r )) ρ ϕ ( f ) − α ϕ,s γ ϕ,s + ε ′ ≤ ϕ ( r ) ρϕ ( f ) − αϕ,sγϕ,s + ε ′ αϕ,s − ε ∗ ≤ ϕ ( r ) ρϕ ( f ) αϕ,s − γ ϕ,s + ε , r ≥ R . (4.2)Case (a) now follows directly from (4.2) and Lemmas 3.1(a) and 4.2(a). Case(b) is more straight forward. (cid:3) Remark 2. If α ϕ,s >
0, it is easy to see that ρ ϕ ( D q f ) ≤ max (cid:26) ρ ϕ ( f ) , ρ ϕ ( f ) α ϕ,s − γ ϕ,s (cid:27) . H. Yu, J. Heittokangas, J. Wang, and Z. T. Wen Proofs of theorems
Proof of Theorem 2.1.
All assertions are true if ρ ϕ ( f ) = ∞ or if α ϕ,s = 0,so we may suppose that ρ ϕ ( f ) < ∞ and α ϕ,s > k ∈ N that ρ ϕ ( D kq f ) ≤ max ( ρ ϕ ( f ) , max ≤ l ≤ k ( ρ ϕ ( f ) α lϕ,s − γ ϕ,s l − X j =0 α jϕ,s )) =: ρ ϕ,k . (5.1)The case k = 1 is obvious by Remark 2. We suppose that (5.1) holds for k ,and we aim to prove (5.1) for k + 1. Applying Remark 2 to the meromorphicfunction D kq f yields ρ ϕ ( D k +1 q f ) = ρ ϕ ( D q ( D kq f )) ≤ max ( ρ ϕ ( D kq f ) , ρ ϕ ( D kq f ) α ϕ,s − γ ϕ,s ) ≤ max ( ρ ϕ ( f ) , max ≤ l ≤ k +1 ( ρ ϕ ( f ) α lϕ,s − γ ϕ,s l − X j =0 α jϕ,s )) = ρ ϕ,k +1 . The assertion (5.1) is now proved. Moreover, it is easy to see that ρ ϕ ( f ) ≤ ρ ϕ,k ≤ ρ ϕ,k +1 for k ∈ N .Suppose first that the coefficients a ( x ) , . . . , a n ( x ) are entire. We divide(2.1) by f ( x ) and make use of (4.2), (5.1) and Lemma 3.1(a) to obtain m ( r, a ) ≤ max ≤ j ≤ n { m ( r, a j ) } + X ≤ j ≤ n m (cid:18) r, D jq ff (cid:19) . max ≤ j ≤ n { m ( r, a j ) } + max ≤ j ≤ n (cid:26) m (cid:18) r, D jq f D j − q f (cid:19)(cid:27) . ϕ ( r ) ρ ϕ ( a ) − ε + ϕ ( r ) ρϕ,n − αϕ,s − γ ϕ,s + ε , r ≥ R . (5.2)Since there exists a sequence { r n } of positive real numbers tending to infinitysuch that m ( r n , a ) ≥ ϕ ( r n ) ρ ϕ ( a ) − ε , we have ρ ϕ ( a ) − ε ≤ ρ ϕ,n − α ϕ,s − γ ϕ,s + ε, where we may let ε → + . This gives us ρ ϕ ( a ) ≤ max ( ρ ϕ ( f ) α ϕ,s − γ ϕ,s , max ≤ l ≤ n − ( ρ ϕ ( f ) α l +1 ϕ,s − γ ϕ,s l X j =0 α jϕ,s )) = max ≤ l ≤ n ( ρ ϕ ( f ) α lϕ,s − γ ϕ,s l − X j =0 α jϕ,s ) , and so α nϕ,s ρ ϕ ( a ) ≤ max ≤ l ≤ n ( α n − lϕ,s ρ ϕ ( f ) − γ ϕ,s l − X j =0 α n − jϕ,s ) ≤ ρ ϕ ( f ) − α nϕ,s γ ϕ,s . Then the assertion (2.5) follows. he ϕ -order for the Askey-Wilson divided differences a ( x ) , . . . , a n ( x ) have poles. Wedivide (2.1) by f ( x ) and make use of (5.1) and Lemma 4.3(a) to obtain N ( r, a ) . max ≤ j ≤ n { T ( r, a j ) } + n X j =0 T ( r, D jq f ) . max ≤ j ≤ n { T ( r, a j ) } + T ( r, f ) + ϕ ( r ) ρϕ,n − αϕ,s − γ ϕ,s + ε + log r . ϕ ( r ) ρ ϕ ( a ) − ε + ϕ ( r ) ρ ϕ ( f )+ ε + ϕ ( r ) ρϕ,n − αϕ,s − γ ϕ,s + ε + log r, r ≥ R . Combining this with (5.2) and noting the fact that f is non-constant, weobtain ρ ϕ ( a ) ≤ max ( ρ ϕ ( f ) , max ≤ l ≤ n ( ρ ϕ ( f ) α lϕ,s − γ ϕ,s l − X j =0 α jϕ,s )) = ρ ϕ,n , and thus, similarly as above, α nϕ,s ρ ϕ ( a ) ≤ max (cid:8) α nϕ,s ρ ϕ ( f ) , ρ ϕ ( f ) − α nϕ,s γ ϕ,s (cid:9) ≤ ρ ϕ ( f ) . Hence the assertion (2.4) follows.(b) Similarly as in Case (a) above, we make use of (5.1) and Lemmas 3.1(b)and 4.3(b) to obtain T ( r, a ) = m ( r, a ) + N ( r, a ) . max ≤ j ≤ n { T ( r, a j ) } + X ≤ j ≤ n m (cid:18) r, D jq ff (cid:19) + n X j =0 T ( r, D jq f ) . ϕ ( r ) ρ ϕ ( a ) − ε + ϕ ( r ) ρ ϕ,n − + ε + log r, r ≥ R . This together with the fact that f is non-constant, we deduce ρ ϕ ( a ) ≤ ρ ϕ,n − ,and so α n − ϕ,s ρ ϕ ( a ) ≤ ρ ϕ ( f ) . This completes the proof. ✷ Proof of Theorem 2.2.
Choose s ( r ) satisfying the assumptions of The-orem 2.1(b). We divide (2.2) by f ( x ) and make use of (5.1) and Lemmas3.1(b) and 4.3(b) to obtain T ( r, a ) . max ≤ j ≤ n +1 { T ( r, a j ) } + X ≤ j ≤ n m (cid:18) r, D jq ff (cid:19) + m (cid:18) r, f (cid:19) + n X j =0 T ( r, D jq f ) . ϕ ( r ) ρ ϕ ( a ) − ε + ϕ ( r ) ρ ϕ,n − + ε + log r, r ≥ R . Similarly as in the proof of Theorem 2.1(b), the assertion follows. ✷ Acknowledgements
The first author would like to thank the support of the China Scholar-ship Council (No. 201806330120). The third author was supported by Na-tional Natural Science Foundation of China (No. 11771090). The fourth au-thor was supported by the National Natural Science Foundation of China(No. 11971288 and No. 11771090) and Shantou University SRFT (NTF18029).4
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Department of Physics and Mathematics, University ofEastern Finland, P.O. Box 111, 80101 Joensuu, Finland
Email address : [email protected], [email protected] (Wang) School of Mathematical Sciences, Fudan University, Shanghai 200433,P.R. China
Email address : [email protected] (Wen) Department of Mathematics, Shantou University, Shantou 515063,Guangdong, P.R. China
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