Normal family of meromorphic functions concerning limited the numbers of zeros
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ORIGINAL RESEARCH PAPER
Normal family of meromorphic functions concerning limited the numbers of zeros Chengxiong Sun1 Received: 15 October 2019 / Accepted: 17 October 2020 Forum D’Analystes, Chennai 2020
Abstract Let k; n 2 N; l 2 Nnf1g; m 2 N [ f0g, and let aðzÞð6 0Þ be a holomorphic function, all zeros of a(z) have multiplicities at most m. Let F be a family of meromorphic functions in D. If for each f 2 F , the zeros of f have multiplicity at least k þ m, and for f 2 F , f l ðf ðkÞ Þn aðzÞ has at most one zero in D, then F is normal in D. Keywords Meromorphic function Normal families Zero numbers
Mathematics Subject Classification Primary 30D35 Secondary 30D45
1 Introduction and main results Let f be a meromorphic function in C and we shall use the usual notations and classical results of Nevanlinna’s theory, such as mðr; f Þ; Nðr; f Þ; Nðr; f Þ; Tðr; f Þ; . . .. Let D be a domain in C and F be a family of meromorphic functions in D. A family F is said to be normal in D, in the sense of Montel, if each sequence fn has a subsequence fnk that converges spherically locally uniformly in D to a meromorphic function or to the constant 1. The following well-known normal conjecture was proposed by Hayman in 1967. Theorem A [1] Let n 2 N, and a 2 Cnf0g. let F be a family of meromorphic function in D. If f n f 0 6¼ a, for each f 2 F , then F is normal in D. This normal conjecture was showed by Yang and Zhang [2] (for n 5), Gu [3] ðforn ¼ 4; 3Þ; Pang [4] (for n 2) and Chen and Fang [5] (for n ¼ 1). For the related results, see Zhang [6], Meng and Hu [7], Deng et al.[8]. & Chengxiong Sun [email protected] 1
Department of Mathematics, Xuanwei No. 9 Senior High School, Yunnan, People’s Republic of China
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C. Sun
Ding et al. [9] studied the general case of f l ðf ðkÞ Þn and and proved the following theorem. Theorem B Let k; l 2 N; n 2 Nnf1g; a 2 Cnf0g. Let F be a family of meromorphic functions in D. If for each f 2 F , the zeros of f have multiplicity at least maxfk; 2g, and for f ; g 2 F , f l ðf ðkÞ Þn and gl ðgðkÞ Þn share a, then F is normal in D. Recently, Meng et al. [10] considered the case of sharing a holomorphic function and and proved the following result. Theorem C Let k; l 2 N; n 2 Nnf1g; m 2 N [ f0g, and let aðzÞð6 0Þ be a holomorphic function, all zeros of a(z) have multiplicities at most m, which is divisible by n þ l. Let F be a family of meromorphic functions in D. If for each f 2 F , the zeros of f have multiplicities at least k þ m þ 1 and all poles of f are of multiplicity at least m þ 1, and for f ; g 2 F , f l ðf ðkÞ Þn and gl ðgðkÞ Þn share a(z), then F is normal in D. By Theorem C, the following question arises naturally: Question 1.1 Is it possible to omit the conditions: (1)‘‘ m is divisible by n þ l’’ and (2)‘‘all poles of f have multiplicity at least m þ 1’’ ? In this paper, we study this problem and obtain the following result. Theorem 1.1 Let k; n 2 N; l 2 Nnf1g; m 2 N [ f0g, and let aðzÞð6 0Þ be a holomorphic function, all
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