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International Journal of Modern Engineering Research (IJM ER) Vo l. 3, Issue. 4, Ju l - Aug. 2013 pp-2197-2204 ISSN: 2249-6645

Further Results On The Basis Of Cauchy’s Proper Bound for the Zeros of Entire Functions of Order Zero Sanjib Kumar Datta1, Manab Biswas 2 1

(Department of Mathematics, University of Kalyani, Kalyani, Dist-Nadia, Pin- 741235, West Bengal, India) 2 (Barabilla High School, P.O. Haptiagach, Dist-Uttar Dinajpur, Pin- 733202, West Bengal, India)

ABSTRACT: A single valued function of one complex variable which is analytic in the finite complex plane is called an entire function . The purpose of this paper is to establish the bounds for the moduli of zeros of entire functions of order zero. Some examples are provided to clear the notions. AMS Subject Classification 2010: Primary 30C15, 30C10, Secondary 26C10.

KEYWORDS AND PHRASES: Zeros of entire functions of order zero, Cauchy’s bound , Proper ring shaped region.

I.

INTRODUCTION, DEFINITIONS AND NOTATIONS

Let, đ?‘ƒ đ?‘§ = đ?‘Ž0 + đ?‘Ž1 đ?‘§ + đ?‘Ž2 đ?‘§ 2 + đ?‘Ž3 đ?‘§ 3 +‌‌+ đ?‘Žđ?‘› −1 đ?‘§ đ?‘›âˆ’1 + đ?‘Žđ?‘› đ?‘§ đ?‘› ; đ?‘Žđ?‘› ≠0 Be a polyno mial of degree n . Datt and Govil [2] ; Govil and Rahaman [5] ; Marden [9] ; Mohammad [10] ; Chattopadhyay, Das, Jain and Konwar [1] ; Joyal, Labelle and Rahaman [6] ; Jain {[7] , [8]}; Sun and Hsieh [11] ; Zilovic, Roytman, Co mbettes and Swamy [13] ; Das and Datta [4] etc. wo rked in the theory of the distribution of the zeros of polynomials and obtained some newly developed results. In this paper we intend to establish some of sharper results concerning the theory of distribution of zeros of entire functions of order zero. The fo llo wing definitions are well known : Definiti on 1 The order đ?œŒ and lo wer order đ?œ† of a mero morphic function f are defined as log đ?‘‡ (đ?‘&#x;,đ?‘“) log đ?‘‡ (đ?‘&#x;,đ?‘“ ) đ?œŒ = limsup and đ?œ† = liminf ∙ log đ?‘&#x; log đ?‘&#x; đ?‘&#x; → ∞ đ?‘&#x;→∞ If f is entire, one can easily verify that log [2]đ?‘€(đ?‘&#x;,đ?‘“ )

log [2]đ?‘€(đ?‘&#x; ,đ?‘“)

đ?œŒ = limsup and đ?œ† = liminf ∙ log đ?‘&#x; đ?‘&#x; → ∞ log đ?‘&#x; đ?‘&#x;→∞ where log k đ?‘Ľ = log log k −1 đ?‘Ľ for đ?‘˜ = 1, 2, 3, ‌.. and log 0 đ?‘Ľ = đ?‘Ľ . If đ?œŒ < ∞ then đ?‘“ is of fin ite order. Also đ?œŒ = 0 means that đ?‘“ is of order zero. In this connection Datta and Biswas [3] gave the following definit ion : Defini tion 2 Let f be a mero morphic function of order zero. Then the quantities đ?œŒ ∗ and đ?œ†âˆ— of f are defined by : đ?‘‡(đ?‘&#x;,đ?‘“ ) đ?‘‡(đ?‘&#x;,đ?‘“ ) đ?œŒ ∗ = limsup and đ?œ†âˆ— = liminf ∙ log đ?‘&#x; đ?‘&#x; → ∞ log đ?‘&#x; đ?‘&#x;→∞ If f is an entire function then clearly đ?œŒ ∗ = limsup đ?‘&#x;→∞

log đ?‘€ đ?‘&#x;,đ?‘“ log đ?‘&#x;

and đ?œ†âˆ— = liminf đ?‘&#x;→∞

log đ?‘€ đ?‘&#x;,đ?‘“ log đ?‘&#x;

∙

II.

LEMMAS

In this section we present a lemma which will be needed in the sequel . ∗ Lemma 1 If đ?‘“(đ?‘§) is an entire function of order đ?œŒ = 0, then for every đ?œ€ > 0 the inequality đ?‘ đ?‘&#x; ≤ log đ?‘&#x; đ?œŒ +đ?œ€ Holds for all sufficiently large đ?‘&#x; where đ?‘ (đ?‘&#x;) is the number of zeros of đ?‘“(đ?‘§) in đ?‘§ ≤ log đ?‘&#x; . Proof. Let us suppose that đ?‘“(đ?‘§) = 1. This supposition can be made without loss of generality because if đ?‘“(đ?‘§) has a zero of đ?‘“ (đ?‘§)

order ′đ?‘šâ€˛ at the origin then we may consider đ?‘” đ?‘§ = đ?‘?. đ?‘š where đ?‘? is so chosen that đ?‘”(0) = 1. Since the function đ?‘”(đ?‘§) and đ?‘§ đ?‘“(đ?‘§) have the same order therefore it will be unimportant for our investigations that the number of zeros of đ?‘”(đ?‘§) and đ?‘“(đ?‘§) differ by đ?‘š. We further assume that đ?‘“(đ?‘§) has no zeros on đ?‘§ = log 2đ?‘&#x; and the zeros đ?‘§đ?‘– ’s of đ?‘“(đ?‘§) in đ?‘§ < log đ?‘&#x; are in non decreasing order of their moduli so that đ?‘§đ?‘– ≤ đ?‘§đ?‘– +1 . Also let đ?œŒ ∗ suppose to be finite where đ?œŒ = 0 is the zero of order of đ?‘“ đ?‘§ . Now we shall make use of Jenson’s formula as state below đ?‘› 2đ?œ‹ đ?‘… 1 log đ?‘“(0) = − log + (1) âˆŤ log đ?‘“(đ?‘… đ?‘’ đ?‘–đ?œ™ ) đ?‘‘đ?œ™ . đ?‘§đ?‘– 2đ?œ‹ đ?‘–=1 0 Let us replace đ?‘… by 2đ?‘&#x; and đ?‘› by đ?‘ (2đ?‘&#x;) in (1) . đ?‘ (2đ?‘&#x;) 2đ?œ‹ 2đ?‘&#x; 1 ∴ log đ?‘“(0) = − log + âˆŤ log đ?‘“(2đ?‘&#x; đ?‘’ đ?‘–đ?œ™ ) đ?‘‘đ?œ™. đ?‘§đ?‘– 2đ?œ‹ đ?‘– =1 0 Since đ?‘“ 0 = 1, ∴ log đ?‘“(0) = log 1 = 0. www.ijmer.co m

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