Deficiency And Relative Deficiency Of E-Valued Meromorphic Functions

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1 Applied Mathematics E-Notes, 323, -8 c ISSN Available free at mirror sites of amen/ Deficiency And Relative Deficiency Of E-Valued Meromorphic Functions Zhaojun Wu, Zuxing Xuan Received 9 May 23 Abstract The purpose of this paper is to discuss the deficiency the relative deficiency of an E-valued meromorphic function f. Some inequalities on the deficiency the relative deficiency of E-valued meromorphic functions are obtained. Results obtained extend some recent results by Wu Xuan. E-Valued Meromorphic Function In this section, we shall introduce some fundamental definitions notations of E- valued meromorphic functions see e.g. [, 2, 6, 7, 8] which will be used in this paper. Let E, be a complex Banach space with Schauder basis {e j } the norm. Then an E-valued function fz defined in C R = { z < R}, < R +, C + = C can be written as fz = f z, f 2 z,..., f k z,... E, where f z, f 2 z,..., f k z,... are the component functions of fz. An E-valued function is called holomorphic meromorphic if all f j z are holomorphic some of f j z are meromorphic. The j-th derivative of fz, where j =, 2,..., is defined by f j z = f j z, f j 2 j z,..., f k z,... We assume that f z = fz. In this paper, we use the letters from the alphabet a, b, c,... to denote the elements of E which are called vectors. The symbol denotes the zero vector of E. We denote vector infinity, complex number infinity, the norm infinity by,, +, respectively. Let fz be an E-valued meromorphic function in C R < R +. A point z C R = { z < r} is called a pole of fz = f z, f 2 z,..., f k z,... if z is a pole Mathematics Subject Classifications: 3D35, 3G3, 46G2. School of Mathematics Statistics, Hubei University of Science Technology, Xianning, Hubei, 437, P. R. China Beijing Key Laboratory of Information Service Engineering, Department of General Education, Beijing Union University, No.97 Bei Si Huan Dong Road, Chaoyang District, Beijing,, P. R. China

2 Z. J. Wu Z. X. Xuan of at least one of the component functions f k z k =, 2,... We denote fz = + when z is a pole. A point z C R is called a zero of fz = f z, f 2 z,..., f k z,... if z is a zero of all the component functions f k z k =, 2,... For any vector a E, A point z C R is called an a-point of fz if only if z a zero of fz a. Let nr, f denote the number of poles of fz in z r, nr, a denote the number of a-points of fz in z r, counting with multiplicities. We define the volume function of fz by V r,, f = V r, f = log r C r ξ log fξ dx dy, ξ = x + iy, V r, a = V r, a, f = log r C r ξ log fξ a dx dy, ξ = x + iy, the counting function of finite or infinite a-points by respectively. Next, we define Nr, f = n, f log r + r r Nr, a = n, a log r + mr, f = mr,, f = mr, a = mr, a, f = = mr, f + Nr, f, nt, f n, f dt t nt, a n, a dt, t log + fre iθ dθ, log + fre iθ a dθ where is called the Nevanlinna characteristic function. Let nr, f denote the number of poles of fz in z r, nr, a denote the number of a-points of fz in z r, ignoring multiplicities. Similarly, we can define the counting function Nr, f Nr, a of nr, f nr, a. Let fz be an E-valued meromorphic function in the whole complex plane. For any vector a E, we define the Nevanlinna deficiency δa = δa, f Θa = Θa, f as following mr, a δa = δa, f = lim inf = V r, a + Nr, a, mr, f Nr, f δ = δ, f = lim inf =, Θa = Θa, f = V r, a + Nr, a

3 2 Deficiency of E-Valued Meromorphic Functions Θ = Θ, f = Nr, f. If δa >, we call a is a deficient vector of f. If fz is an E-valued meromorphic function in the whole complex plane, then the order the lower order of fz are defined by λf = r log log r µf = lim inf r log. log r We call the E-valued meromorphic function f admissible if 2 Lemmas log r = +. In this section, we shall introduce some fundamental results which will be used in the proof of the main results of this paper. For convenience, we give the following definitions. DEFINITION 2. [2]. For an E-valued meromorphic function fz, we denote by Sr, f any quantity such that or Sr, f = Olog + log r, r + Sr, f = o, r + without restriction if fz is of finite order otherwise except possibly for a set of values of r of finite linear measure. DEFINITION 2.2 [2] Let E n be an n-dimensional projective space of E with a basis {e j } n. The projective operator P n : E E n is a realization of E n associated to the basis. An E-valued meromorphic function fz in C R < R + is of compact projection, if for any given ε >, P n fz fz < ε has suffi ciently large n in any fixed compact subset D C R. LEMMA 2.3 [2]. Let fz be a nonconstant E-valued meromorphic function in C R. Then for < r < R, a E, fz a, = V r, a + Nr, a + mr, a + log + c q a + εr, a. Here εr, a is a function such that εr, a log + a + log 2, εr, c q a E is the coeffi cient of the first term in the Laurent series at the point a.

4 Z. J. Wu Z. X. Xuan 3 LEMMA 2.4 [2]. Let fz be a nonconstant E-valued meromorphic function of compact projection in C a [k] E k =, 2,..., q be q 3 distinct points. Then for < r < R, q 2 + Gr, f q + Gr, f [V r, a [k] + Nr, a [k] ] + Sr, f k= [V r, a [k] + Nr, a [k] ] + Nr, f + Sr, f k= where Gr, f = r dt log fξ dx dy, ξ = x + iy. C r LEMMA 2.5 [6]. If an E-valued meromorphic function fz in C is of compact projection, then for a positive integer k, we have Sr, f k = Sr, f log + f k re iθ fre iθ dθ = Sr, f. 3 Deficiency of E-Valued Meromorphic Function In 992, Lahiri [3] proved the following theorem on vector valued transcendental integral function connecting the deficiencies of the function with its derivative. THEOREM A. Let fz = f z, f 2 z,..., f n z be a vector-valued transcendental integral function of finite order. Then a C n δa δ, f. Extending Theorem A recently, Wu Xuan [6] prove the following result. THEOREM B. Let fz be a finite order admissible E-valued meromorphic function of compact projection in C assume δ =. Then δa δ, f. We extend the above result to the higher derivatives as follows.

5 4 Deficiency of E-Valued Meromorphic Functions THEOREM 3.. Let fz be a finite order admissible E-valued meromorphic function of compact projection in C assume δ =. Then for any positive integer k, we have δa δ, f k. PROOF. For any q 2 vectors {a [µ] } in E, put We can get F z = j= log + F re iθ dθ m r,, f k + By Wu Xuan [6], we have So that by 2, we can get log + F re iθ dθ mr, a [µ] m r,, f k + Thus by Lemma 2.5, we have It follows from Lemma 2.3 that m r,, f k + N Thus from 3 4, we deduce mr, a [µ] + N Therefore, we have fz a [j]. log + {F re iθ f k re iθ }dθ. mr, a [µ] log + 2q δ. 2 log + {F re iθ f k re iθ }dθ + log + 2q δ. mr, a [µ] m r,, f k + Sr, f. 3 r,, f k + V r,, f k = T r, f k + O. 4 r,, f k + V r,, f k T r, f k + Sr, f. N r,, f k + V r,, f k + T r, f k T r, f k mr, a [µ] o. 5

6 Z. J. Wu Z. X. Xuan 5 Now, basic estimates in E-valued Nevanlinna theory yield T r, f k = mr, f k + Nr, f k = mr, f + Nr, f k + mr, f + Nr, f + knr, f + + knr, f + log + f k re iθ fre iθ dθ log + f k re iθ fre iθ dθ log + f k re iθ fre iθ dθ. Thus we can get from the above inequality Lemma 2.5 that Hence from 5 6, we can get T r, f k N r,, f k + V r,, f k + T r, f k k + kδ. 6 T r, f k mr, a [µ] o N r,, f k + V r,, f k + lim inf T r, f k T r, f k mr, a [µ] N r,, f k + V r,, f k + lim inf lim inf T r, f k T r, f k N r,, f k + V r,, f k δ a [j] j= + T r, f k k + kδ. o mr, a [µ] Since q > was arbitrary δ =, we have δa δ, f k. From Theorem 3., we have COROLLARY 3.2. Let fz be a finite order admissible E-valued meromorphic function of compact projection in C assume δ =. If fz has at least one deficient vector a E, then for any positive integer k, the vector is the deficient vector of f k.

7 6 Deficiency of E-Valued Meromorphic Functions COROLLARY 3.3. Let fz be a finite order admissible E-valued meromorphic function of compact projection in C assume δa = δ =. Then for any positive integer k, δ, f k =. EXAMPLE 3.4. Put fz = e z, e z,..., e z,... Then f j z = e z, e z,..., e z,..., j =, 2,.... For any non-zero vector a E, we have δa =, δ =, δ = δ, f k = holds for any positive integer k. Thus δa δ, f k. 4 Relative Deficiency The concept of the relative Nevanlinna defect of meromorphic function was due to Milloux [4] Xiong Qinglai also Hiong Qinglai [5]. Lahiri [3] extended this concept to meromorphic vector valued function in 99. Most recently, Wu Xuan [6] considered the relative deficiency of E-valued meromorphic function gave the following definition theorem. DEFINITION 4.. If k is a positive integer then the number Θ k a, f = V r, a, f k + Nr, a, f k is called the relative deficiency of the value a E with respect to distinct zeros. THEOREM C. Let fz be an admissible E-valued meromorphic function of compact projection in C let a [µ] µ =, 2,, p b [λ] λ =, 2,, q, q 2, be elements of E, distinct within each set. Then for all positive integers k, Θ k b [λ], f + q 2 λ= p δa [µ], f q. In this section, we continue to study the relative deficiency of E-valued meromorphic function prove the following theorems. THEOREM 4.2. Let fz be a finite order admissible E-valued meromorphic function of compact projection in C let a, b, c be three distinct elements in E { } with a. Then δa, f + Θ k b, f + Θ k c, f 2.

8 Z. J. Wu Z. X. Xuan 7 PROOF. By Lemma 2.5, we get that mr, a, f = mr,, f k + log + f k re iθ fre iθ a f k re iθ dθ log + = mr,, f k + Sr, f. Now by Lemma , we can get T r, f k Nr,, f k + V r,, f k f k re iθ fre iθ a dθ + Nr, a, f + V r, a, f + Sr, f Nr, a, f + V r, a, f Nr,, f k + V r,, f k + Nr,, f k + V r,, f k + Nr, b, f k + V r, b, f k + Nr, c, f k + V r, c, f k + Sr, f Nr, a, f + V r, a, f + Nr, b, f k + V r, b, f k + Nr, c, f k + V r, c, f k + Sr, f. Dividing both sides of the above inequality by taking limit superior we have i. e. or Nr, a, f + V r, a, f Nr, c, f k + V r, c, f k Nr, b, f k + V r, b, f k Sr, f, δa, f + Θ k b, f + Θ k b, f +, δa, f + Θ k b, f + Θ k c, f 2. THEOREM 4.3. Let a be any vector in E f be a finite order admissible E-valued meromorphic function of compact projection in C. Then Θ, f + Θa, f + Θ k, f 2. PROOF. By Lemma 2.4, we have Nr,, f + V r,, f + Nr, a, f + V r, a, f + Nr, f + Sr, f Nr,, f + V r,, f + Nr, a, f + V r, a, f + Nr, f k + Sr, f.

9 8 Deficiency of E-Valued Meromorphic Functions Dividing both sides of the above inequality by taking limit superior we have i. e. or Nr,, f + V r,, f Nr, f k Sr, f, Nr, a, f + V r, a, f Θ, f + Θa, f + Θ k, f +, Θ, f + Θa, f + Θ k, f 2. Acknowledgment. This research was partly supported by the National Natural Science Foundation of China Grant No by the Science Foundation of Educational Commission of Hubei Province Grant no. Q2328. References [] C. G. Hu, The Nevanlinna theory its related Paley problems with applications in infinite-dimensional spaces, Complex Var. Elliptic Equ., 562, [2] C. G. Hu Q. J. Hu, The Nevanlinna s theorems for a class, Complex Var. Elliptic Equ., 526, [3] I. Lahiri, Milloux theorem deficiency of vector-valued meromorphic functions, J. Indian Math. Soc., 599, [4] H. Milloux, Les dérivées des fonctions méromorphes et la théorie des défauts French, Ann. Sci. École Norm. Sup., [5] Q. L. Xiong, A fundamental inequality in the theory of meromorphic functions its applications, Chinese Mathematics, 9967, [6] Z. J. Wu Z. X. Xuan, Deficiency of E-valued meromorphic functions, Bull. Belg. Math. Soc. Simon Stevin, 922, [7] Z. J. Wu Z. X. Xuan, Characteristic functions Borel exceptional values of E-valued meromorphic functions, Abstr. Appl. Anal., 22, Art. ID 2656, 5 pp. [8] Z. X. Xuan N. Wu, On the Nevanlinna s theory for vector-valued mappings, Abstr. Appl. Anal., 2, Art. ID , 5 pp.