A note on partial sharing of values of meromorphic functions with their shifts

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A note on partial sharing of values of meromorphic functions with their shifts ✩ K.S. Charak a , R.J. Korhonen b,∗ , Gaurav Kumar a a

Department of Mathematics, University of Jammu, Jammu 180 006, India Department of Physics and Mathematics, University of Eastern Finland, P.O. Box 111, FI-80101 Joensuu, Finland b

a r t i c l e

i n f o

Article history: Received 12 August 2015 Available online xxxx Submitted by V. Andrievskii Keywords: Meromorphic functions Partial sharing of values Shift Difference operators Nevanlinna theory

a b s t r a c t We introduce the notion of partial value sharing as follows. Let E(a, f ) be the set of zeros of f (z) − a(z), where each zero is counted only once and a is a meromorphic function, small with respect to f . A meromorphic function f is said to share a partially with a meromorphic function g if E(a, f ) ⊆ E(a, g). We show that partial value sharing of f (z) and f (z + c) involving 3 or 4 values combined with an appropriate deficiency assumption is enough to guarantee that f (z) ≡ f (z + c), provided that f (z) is a meromorphic function of hyper-order strictly less than one and c ∈ C. © 2015 Elsevier Inc. All rights reserved.

1. Introduction The famous five point theorem due to R. Nevanlinna [12] says that if two non-constant meromorphic functions f and g share five distinct values ignoring multiplicities (IM), then f (z) ≡ g(z). The beauty of this result lies in the fact that there is no counterpart of this result in the real function theory. Another famous result in this direction is the four point theorem by R. Nevanlinna [12] which states that if two distinct non-constant meromorphic functions f and g share four distinct values counting multiplicities (CM), then f = T ◦ g, where T is a Möbius map. These results initiated the study of uniqueness of meromorphic functions. The study becomes more interesting if the function g is related to f . Investigations began with the sharing of values by f and f  [5]. Korhonen and Halburd [6,7] and, independently, Chiang and Feng [2,3] developed a parallel difference version to the usual Nevanlinna theory for meromorphic functions of finite order and this gives the direction to study the uniqueness of f (z) and its shift f (z + c), where c ∈ C \ {0}. ✩ The second author was supported in part by the Academy of Finland grants 286877 and 268009. The third author is supported by the University Grants Commission of India as Junior Research Fellow. * Corresponding author. E-mail addresses: kscharak7@rediffmail.com (K.S. Charak), risto.korhonen@uef.fi (R.J. Korhonen), [email protected] (G. Kumar).

http://dx.doi.org/10.1016/j.jmaa.2015.10.069 0022-247X/© 2015 Elsevier Inc. All rights reserved.

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Key results of this difference version of Nevanlinna theory were extended by Halburd, Korhonen and Tohge [8] to meromorphic functions of hyper-order strictly less than one. Throughout this paper, we only consider such meromorphic functions which are non-constant and meromorphic in the whole of C. For such a meromorphic function f , a meromorphic function ω such that T (r, ω) = o(T (r, f )), where r → ∞ outside of a possible exceptional set of finite logarithmic measure, is ˆ ) = S(f ) ∪{∞}. called a small function of f . The family of all small functions of f is denoted by S(f ) and S(f Recently J. Heittokangas et al. [11,10] considered the problem of value sharing for shifts of meromorphic functions. Precisely, they proved the following results: Theorem 1.1. (See [10, Theorem 2.1(a)].) Let f be a meromorphic function of finite order and let c ∈ C \{0}. ˆ ) with period c CM, then f (z) = If f (z) and f (z + c) share three distinct periodic functions a1 , a2 , a3 ∈ S(f f (z + c) for all z ∈ C. Theorem 1.1 is improved by replacing “sharing three small functions CM” by “2 CM + 1 IM” as: Theorem 1.2. (See [11, Theorem 2].) Let f be a meromorphic function of finite order, let c ∈ C \ {0}, and ˆ ) be three distinct periodic functions with period c. If f (z) and f (z + c) share a1 , a2 CM let a1 , a2 , a3 ∈ S(f and a3 IM, then f (z) = f (z + c) for all z ∈ C. The cases “1 CM + 2 IM” and “3 IM” are left open in [11]. In the present paper, we make certain investigations about the open situations “1 CM + 2 IM” and “3 IM”. For basic terms and notations of the value distribution theory of Nevanlinna one may refer to [9,1]. Example 1.3. Consider f (z) =

(1+cos z)2 (1−cos z) . e2iz

Then, we have the following observations:

(i) For c = π, f (z) and f (z + c) share 0, −1 IM and ∞ CM. (ii) δ (a, f ) = 0 for all a ∈ C. (iii) δ(∞, f ) = 1. Proofs of (i) and (iii) are immediate. We prove (ii) here. After some simple calculations, one can easily deduce that f (z) = R(z) ◦ eiz , where R(z) =

z 6 + 2z 5 − z 4 − 4z 3 − z 2 + 2z + 1 . −8z 5

The only poles of R(z) are at 0 (with multiplicity 5) and at ∞ (with multiplicity 1). Thus, for each a ∈ C, R−1 (a) contains six nonzero complex numbers counted according to their multiplicity. Let a ∈ C. Let R−1 (a) = {bk : k = 1, 2, 3, 4, 5, 6} (each bk ∈ C \ {0}). Then,

N

 r,

1 f −a

 =

6  k=1

 N

r,

1 eiz − bk

 = 6T (r, ez ).

  1 Thus N r, f −a is same for every a ∈ C. Thus, if a is a deficient value, that is δ(a, f ) > 0, then each value in C is a deficient value which contradicts the fact that set of deficient values of any nonconstant meromorphic function is at most countable. Thus δ(a, f ) = 0 for all a ∈ C. ˆ ) partially with a meromorphic function Definition 1.4. We say that a meromorphic function f shares a ∈ S(f g if E(a, f ) ⊆ E(a, g), where E(a, f ) is the set of zeros of f (z) − a(z), where each zero is counted only once.

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ˆ ) ∩ S(g). ˆ Let f and g be two nonconstant distinct meromorphic functions and a(z) ∈ S(f We denote by N 0 (r, a, f, g) the counting function of common solutions of f (z) −a(z) = 0 and g(z) −a(z) = 0, each counted only once. Put  N 12 (r, a, f, g) = N

1 r, f −a



 +N

1 r, g−a

 − 2N 0 (r, a, f, g).

That is, N 12 (r, a, f, g) denotes the counting function of distinct solutions of the simultaneous equations f (z) − a(z) = 0 and g(z) − a(z) = 0. Note that in Example 1.3, f (z) and f (z + c) share ∞ CM and 0, −1 IM but f (z) ≡ f (z + c). Thus the case 1 CM + 2 IM (and hence 3 IM) does not hold in general. But under certain sharp conditions, we have obtained some uniqueness results for shifts. Theorem 1.5. Let f be a meromorphic function of hyper-order < 1 and c ∈ C \ {0}. Let a1 , a2 , a3 ∈ ˆ ) be distinct periodic functions with period c such that f (z) share a1 , a2 , a3 partially with f (z + c) and S(f 3 k=1 δ(ak , f ) > 1. Then f (z) ≡ f (z + c). 3 be dropped. For, consider Example 1.3, f (z) shares a1 = 0, The condition k=1 δ(ak , f ) > 1 cannot 3 a2 = −1 and a3 = ∞ with f (z + c) and k=1 δ(ak , f ) = 1 but f (z) ≡ f (z + c). The number “3” associated with the condition in Theorem 1.5 is sharp. For, consider f (z) = ez , and 2 f (z + c) = −ez for c = πi. Then f (z) and f (z + c) share a1 = 0 and a2 = ∞ and k=1 δ(ak , f ) = 2 > 1 but f (z) ≡ f (z + c). ˆ ) Theorem 1.6. Let f be a meromorphic function of hyper-order < 1 and c ∈ C \ {0}. Let a1 , a2 , a3 , a4 ∈ S(f ˆ be such that f (z) share a1 , a2 , a3 , a4 partially with f (z + c) and δ(a, f ) > 0 for some a ∈ S(f ). Then f (z) ≡ f (z + c). The number four associated with the given condition in the hypothesis is sharp. For, recall Example 1.3, f (z) and f (z + c) share 0, −1, ∞ and δ(∞, f (z)) = 1 > 0 but f (z) ≡ f (z + c). It still remains open whether ˆ ). However, if f is entire it follows immediately by we can drop the condition δ(a, f ) > 0 for some a ∈ S(f Theorem 1.6 that conditions on deficiencies are no longer required. Corollary 1.7. Let f be an entire function of hyper-order < 1 and c ∈ C \ {0}. Let a1 , a2 , a3 ∈ C be such that f (z) share a1 , a2 , a3 , partially with f (z + c). Then f (z) ≡ f (z + c). Remark 1.8. If f and g are two distinct nonconstant meromorphic functions sharing four values IM, then f and g must be transcendental (see [4, Theorem G]). Thus, if f is a nonconstant meromorphic function of hyper-order < 1 such that f (z) and f (z + c) share four distinct values IM, where c ∈ C \ {0} with Δc f ≡ 0, then f assumes every value in C∞ infinitely often. Here the difference operator is defined by Δc f (z) := f (z + c) − f (z). The deficiency assumption of Theorem 1.6 may be replaced by assuming that there is a fifth small function a involved in the value sharing of f and its shift. In this case it is enough to assume that the counting function N 0 (r, a, f (z), f (z + c)) is not small. Theorem 1.9. Let f be a meromorphic function of hyper-order < 1 and c ∈ C\{0}. Let a1 , a2 , a3 , a4 be distinct ˆ ) such that f (z) share a1 , a2 , a3 , a4 partially with f (z +c). If N 0 (r, a, f (z), f (z +c)) = S(r, f ) elements of S(f ˆ ) \ {a1 , a2 , a3 , a4 }, then f (z) ≡ f (z + c). for some a ∈ S(f

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2. Proofs of main results First we recall the following results for use in the proof of main results. The first lemma is applicable to estimate various types of shifted counting functions, and the characteristic function. Lemma A. (See [8, Lemma 8.3].) Let T : [0, +∞) → [0, +∞) be a non-decreasing continuous function, and let s ∈ (0, +∞). If the hyper-order of T is strictly less than one, i.e., lim sup r→∞

log log T (r) = ς < 1, log r

then  T (r + s) = T (r) + o

T (r)



r1−ς−ε

,

where ε > 0 and r → ∞ outside of a set of finite logarithmic measure. The first difference analogue of the second main theorem was proved in [7]. The following one-dimensional reduction of [8, Theorem 2.1] is its extension to the case hyper-order < 1. Theorem B. (See [8, Theorem 2.1].) Let c ∈ C, and let f be a meromorphic function of hyper-order < 1 such that Δc f ≡ 0. Let q ≥ 2, and let a1 (z), . . . , aq (z) be distinct meromorphic periodic small functions of f with period c. Then m(r, f ) +

q 

 m r,

1 f − ak

k=1

 ≤ 2T (r, f ) − Npair (r, f ) + S(r, f )

where  Npair (r, f ) := 2N (r, f ) − N (r, Δc f ) + N

1 r, Δc f



and the exceptional set associated with S(r, f ) is of finite logarithmic measure. Lemma C. (See [14, Lemma 4].) Let f and g be nonconstant meromorphic functions and let a1 , a2 , a3 , a4 , a5 ˆ ) ∩ S(g). ˆ be five distinct elements in S(f If f ≡

g, then N 0 (r, a5 , f, g) ≤

4 

N 12 (r, ak , f, g) + S(r, f ) + S(r, g).

k=1

Proof of Theorem 1.5. Suppose on the contrary that f (z) ≡ f (z + c). First, suppose that a1 , a2 , a3 ∈ S(f ). Since f (z) shares a1 , a2 , a3 partially with f (z + c), we have 3 

 N

k=1

r,

1 f − ak

 ≤N

  1 r, . Δc f

(2.1)

By the second main theorem for small functions [9], we have T (r, f ) ≤

3  k=1

 N

r,

1 f − ak

 + S(r, f ).

(2.2)

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By (2.1) and (2.2), we get  T (r, f ) ≤ N

r,

1 Δc f

 + S(r, f ).

(2.3)

By Lemma A, we have N (r, f (z + c)) ≤ N (r + |c|, f ) = N (r, f ) + S(r, f ), and so N (r, Δc f ) ≤ 2N (r, f ) + S(r, f ). Hence from Theorem B and (2.3), it follows that

T (r, f ) ≤

3 









1 r, f − ak

N

k=1

≤

3 

1 r, f − ak

N

k=1

≤

3  k=1

 N

r,

1 f − ak

   1 − 2N (r, f ) + N r, − N (r, Δc f ) + S(r, f ) Δc f 

1 r, Δc f

−N

 + S(r, f )

 − T (r, f ) + S(r, f ).

That is,

2T (r, f ) ≤

3 

 N

k=1

r,

1 f − ak

 + S(r, f ),

and so 2≤3−

3 

δ(ak , f ).

k=1

Hence, 3 

δ(ak , f ) ≤ 1,

k=1

a contradiction. On the other hand, if one of ak ’s is ∞, say a3 = ∞, then we take bk (z) = 1/[ak (z) − a(z)], k = 1, 2 and b3 = 0, where a(z) ∈ S(f ) \ {a1 (z), a2 (z)}. Define F (z) = 1/[f (z) − a(z)]. Then F (z) is a meromorphic function of hyper-order strictly less than one with F (z) sharing b1 , b2 , b3 partially with F (z + c). Then by 3 3 the above case, k=1 δ(bk , F ) ≤ 1 implies k=1 δ(ak , f ) ≤ 1, a contradiction. 2 In order to prove Theorem 1.6 we shall first prove the following lemma which is a generalisation of Theorem F in [4]: Lemma 2.1. Let f be a meromorphic function of hyper-order < 1 and let c ∈ C \ {0} be such that Δc f ≡ 0. ˆ ) be distinct, and such that f (z) shares a1 , a2 , a3 , a4 partially with f (z + c). Then Let a1 , a2 , a3 , a4 ∈ S(f   1 N r, f −a = 2T (r, f ) + S(r, f ); k   1 (ii) N r, f −c + S(r, f ) = T (r, f ), c = ak , k = 1, 2, 3, 4. (i)

4

k=1

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Proof. (i) We may assume that ak ∈ S(f ) for all k = 1, 2, 3, 4. Then, since f (z) shares a1 , a2 , a3 , a4 partially with f (z + c), we have 4 

 N

r,

k=1

1 f − ak

 ≤N

  1 r, . Δc f

Using this inequality and Yamanoi’s second main theorem for small functions [13], we get  1 + S(r, f ) f − ak k=1   1 + S(r, f ) ≤ N r, Δc f

2T (r, f ) + S(r, f ) ≤

4 



N

r,

≤ T (r, Δc f ) + S(r, f ) ≤ T (r, f (z + c)) + T (r, f ) + S(r, f ) = 2T (r, f ) + S(r, f ). Thus, 4 

 N

1 f − ak

r,

k=1

 = 2T (r, f ) + S(r, f )

which proves (i). (ii) Let c = ak , k = 1, 2, 3, 4. Then again by Yamanoi’s second main theorem [13], and by (i), we find that  3T (r, f ) ≤ N

1 f −c

r, 

1 r, f −c

=N

 +

4  k=1



N

 r,

1 f − ak

 + S(r, f )

+ 2T (r, f ) + S(r, f ).

  1 That is, T (r, f ) ≤ N r, f −c + S(r, f ) ≤ T (r, f ), which implies that  N

1 r, f −c

 + S(r, f ) = T (r, f ).

2

Proof of Theorem 1.6. Suppose on the contrary that f (z) ≡ f (z + c). We may suppose that each ak ∈ C, k = 1, 2, 3, 4, since otherwise if any one of ak is ∞, we may proceed as in the proof of Theorem 1.5. Now since f (z) shares a1 , a2 , a3 , a4 partially with f (z + c), we have 4  k=1

 N

1 r, f − ak



  1 ≤ N r, . Δc f

And, then Yamanoi’s second main theorem gives  1 + S(r, f ) 2T (r, f ) ≤ f − ak k=1   1 ≤ N r, + S(r, f ). Δc f 4 

 N r,

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Now, by Theorem B, we arrive at 2T (r, f ) ≤

4  k=1

N

 r,

1 f − ak



 − 2N (r, f ) − N

1 Δc f

r,

 + N (r, Δc f ) + S(r, f ).

(2.4)

Since N (r, Δc f ) ≤ N (r, f (z + c)) + N (r, f ) and, by Lemma A, N (r, f (z + c)) = N (r, f ) + S(r, f ), it follows from (2.4) that 2T (r, f ) ≤

4 

 N

k=1

≤

4 

r,

1 f − ak



1 r, f − ak

N

k=1



 −N

r,

1 Δc f

 + S(r, f )

 − 2T (r, f ) + S(r, f ).

That is, 4T (r, f ) ≤

4 

 N

r,

k=1

1 f − ak

 + S(r, f ) ≤ 4T (r, f )

and so, 4T (r, f ) =

4 

 N

r,

k=1

1 f − ak

 + S(r, f ).

  1 Hence, T (r, f ) = N r, f −a + S(r, f ) for each k ∈ {1, 2, 3, 4}. This implies that δ(ak , f ) = 0 for each k k ∈ {1, 2, 3, 4}. Also by Lemma 2.1, since E(ak , f (z)) ⊆ E(ak , f (z + c)) for all k = 1, 2, 3, 4, we have that δ(b, f ) = 0 for ˆ ) \ {a1 , a2 , a3 , a4 }, and hence δ(a, f ) = 0 for all a ∈ S(f ˆ ), a contradiction. 2 all b ∈ S(f Proof of Theorem 1.9. Suppose on the contrary that f (z) ≡ f (z +c). Since f (z) shares a1 , a2 , a3 , a4 partially with f (z + c), E(ak , f (z)) ⊆ E(ak , f (z + c)), k = 1, 2, 3, 4, and so  N 0 (r, ak , f (z), f (z + c)) = N

r,

1 f − ak



for k = 1, 2, 3, 4. Thus, by Lemma A, we have  N 12 (r, ak , f (z), f (z + c)) = N

r,

1 f (z + c) − ak

 −N

 r,

1 f (z) − ak



= S(r, f ), k = 1, 2, 3, 4. ˆ ) \ {a1 , a2 , a3 , a4 }. Then using (2.5) in Lemma C with g(z) replaced by f (z + c), we have Let a ∈ S(f N 0 (r, a, f (z), f (z + c)) ≤

4 

N 12 (r, ak , f (z), f (z + c)) + S(r, f (z)) + S(r, f (z + c))

k=1

= S(r, f ), which contradicts the assumptions of the theorem. 2

(2.5)

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References [1] W. Cherry, Z. Ye, Nevanlinna’s Theory of Value Distribution: The Second Main Theorem and Its Error Terms, Springer Monogr. Math., Springer-Verlag, Berlin, 2001. [2] Y.M. Chiang, S.J. Feng, On the Nevanlinna characteristic of f (z + η) and difference equations in the complex plane, Ramanujan J. 16 (1) (2008) 105–129, http://dx.doi.org/10.1007/s11139-007-9101-1. [3] Y.M. Chiang, S.J. Feng, On the growth of logarithmic differences, difference quotients and logarithmic derivatives of meromorphic functions, Trans. Amer. Math. Soc. 361 (7) (2009) 3767–3791, http://dx.doi.org/10.1090/S0002-9947-09-04663-7. [4] G.G. Gundersen, Meromorphic functions that share four values, Trans. Amer. Math. Soc. 277 (2) (1983) 545–567, http:// dx.doi.org/10.2307/1999223. [5] G.G. Gundersen, Meromorphic functions that share two finite values with their derivative, Pacific J. Math. 105 (2) (1983) 299–309, http://projecteuclid.org/euclid.pjm/1102723331. [6] R.G. Halburd, R.J. Korhonen, Difference analogue of the lemma on the logarithmic derivative with applications to difference equations, J. Math. Anal. Appl. 314 (2) (2006) 477–487, http://dx.doi.org/10.1016/j.jmaa.2005.04.010. [7] R.G. Halburd, R.J. Korhonen, Nevanlinna theory for the difference operator, Ann. Acad. Sci. Fenn. Math. 31 (2) (2006) 463–478. [8] R.G. Halburd, R.J. Korhonen, K. Tohge, Holomorphic curves with shift-invariant hyperplane preimages, Trans. Amer. Math. Soc. 366 (8) (2014) 4267–4298, http://dx.doi.org/10.1090/S0002-9947-2014-05949-7. [9] W.K. Hayman, Meromorphic Functions, Oxford Math. Monogr., Clarendon Press, Oxford, 1964. [10] J. Heittokangas, R. Korhonen, I. Laine, J. Rieppo, Uniqueness of meromorphic functions sharing values with their shifts, Complex Var. Elliptic Equ. 56 (1–4) (2011) 81–92, http://dx.doi.org/10.1080/17476930903394770. [11] J. Heittokangas, R. Korhonen, I. Laine, J. Rieppo, J. Zhang, Value sharing results for shifts of meromorphic functions, and sufficient conditions for periodicity, J. Math. Anal. Appl. 355 (1) (2009) 352–363, http://dx.doi.org/10.1016/ j.jmaa.2009.01.053. [12] R. Nevanlinna, Le théorème de Picard–Borel et la théorie des fonctions méromorphes, in: Collections de monographies sur la théorie des fonctions, Gauthier-Villars, Paris, 1929, VII + 174 pp. [13] K. Yamanoi, The second main theorem for small functions and related problems, Acta Math. 192 (2) (2004) 225–294, http://dx.doi.org/10.1007/BF02392741. [14] H.-X. Yi, On one problem of uniqueness of meromorphic functions concerning small functions, Proc. Amer. Math. Soc. 130 (6) (2002) 1689–1697, http://dx.doi.org/10.1090/S0002-9939-01-06245-1 (electronic).