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International Journal of Modern Engineering Research (IJMER) Vol. 3, Issue. 5, Sep - Oct. 2013 pp-2606-2614 ISSN: 2249-6645
A Special Type Of Differential Polynomial And Its Comparative Growth Properties Sanjib Kumar Datta1, Ritam Biswas2 1
2
Department of Mathematics, University of Kalyani, Kalyani, Dist.- Nadia, PIN- 741235, West Bengal, India. Murshidabad College of Engineering and Technology, Banjetia, Berhampore, P. O.-Cossimbazar Raj, PIN-742102, West Bengal, India
Abstract: Some new results depending upon the comparative growth rates of composite entire and meromorphic function and a special type of differential polynomial as considered by Bhooshnurmath and Prasad[3] and generated by one of the factors of the composition are obtained in this paper. AMS Subject Classification (2010) : 30D30, 30D35.
Keywords and phrases: Order (lower order), hyper order (hyper lower order), growth rate, entire function, meromorphic function and special type of differential polynomial.
I.
INTRODUCTION, DEFINITIONS AND NOTATIONS.
Let â„‚ be the set of all finite complex numbers. Also let f be a meromorphic function and g be an entire function defined on â„‚ . In the sequel we use the following two notations: đ?‘– log [đ?‘˜] đ?‘Ľ = log(log [đ?‘˜âˆ’1] đ?‘Ľ) đ?‘“đ?‘œđ?‘&#x; đ?‘˜ = 1,2,3, ‌ ; log [0] đ?‘Ľ = đ?‘Ľ and đ?‘–đ?‘– exp[đ?‘˜] đ?‘Ľ = exp đ?‘’đ?‘Ľđ?‘? đ?‘˜ −1 đ?‘Ľ đ?‘“đ?‘œđ?‘&#x; đ?‘˜ = 1,2,3, ‌ ; exp[0] đ?‘Ľ = đ?‘Ľ. The following definitions are frequently used in this paper: Definition 1 The order đ?œŒđ?‘“ and lower order đ?œ†đ?‘“ of a meromorphic function f are defined as lim â Ąsup log đ?‘‡(đ?‘&#x;, đ?‘“) đ?œŒđ?‘“ = đ?‘&#x; →∞ log đ?‘&#x; and log đ?‘‡(đ?‘&#x;, đ?‘“) inf đ?œ†đ?‘“ = limđ?‘&#x;â Ąâ†’âˆž . log đ?‘&#x; If f is entire, one can easily verify that [2] đ?‘€(đ?‘&#x;, đ?‘“) lim â Ąsup log đ?œŒđ?‘“ = đ?‘&#x;→∞ log đ?‘&#x; and log [2] đ?‘€(đ?‘&#x;, đ?‘“) â Ąinf đ?œ†đ?‘“ = limđ?‘&#x;→∞ . log đ?‘&#x; Definition 2 The hyper order đ?œŒđ?‘“ and hyper lower order đ?œ†đ?‘“ of a meromorphic function f are defined as follows đ?œŒđ?‘“ =
lim â Ąsup đ?‘&#x; →∞
đ?œ†đ?‘“ =
lim â Ąinf đ?‘&#x;→∞
đ?œŒđ?‘“ =
lim â Ąsup đ?‘&#x; →∞
log [2] đ?‘‡(đ?‘&#x;, đ?‘“) log đ?‘&#x;
and log [2] đ?‘‡(đ?‘&#x;, đ?‘“) . log đ?‘&#x;
If f is entire, then log [3] đ?‘€(đ?‘&#x;, đ?‘“) log đ?‘&#x;
and log [3] đ?&#x2018;&#x20AC;(đ?&#x2018;&#x;, đ?&#x2018;&#x201C;) . log đ?&#x2018;&#x; Definition 3 The type đ?&#x153;?đ?&#x2018;&#x201C; of a meromorphic function f is defined as follows lim â Ąsup đ?&#x2018;&#x2021;(đ?&#x2018;&#x;, đ?&#x2018;&#x201C;) đ?&#x153;?đ?&#x2018;&#x201C; = đ?&#x2018;&#x;â&#x2020;&#x2019;â&#x2C6;&#x17E; , 0 < đ?&#x153;&#x152;đ?&#x2018;&#x201C; < â&#x2C6;&#x17E;. đ?&#x2018;&#x;đ?&#x153;&#x152;đ?&#x2018;&#x201C; When f is entire, then lim â Ąsup log đ?&#x2018;&#x20AC;(đ?&#x2018;&#x;, đ?&#x2018;&#x201C;) đ?&#x153;?đ?&#x2018;&#x201C; = đ?&#x2018;&#x; â&#x2020;&#x2019;â&#x2C6;&#x17E; , 0 < đ?&#x153;&#x152;đ?&#x2018;&#x201C; < â&#x2C6;&#x17E;. đ?&#x2018;&#x;đ?&#x153;&#x152;đ?&#x2018;&#x201C; Definition 4 A function đ?&#x153;&#x2020;đ?&#x2018;&#x201C; (đ?&#x2018;&#x;) is called a lower proximate order of a meromorphic function f of finite lower order đ?&#x153;&#x2020;đ?&#x2018;&#x201C; if đ?&#x153;&#x2020;đ?&#x2018;&#x201C; =
(i) (ii)
lim â Ąinf đ?&#x2018;&#x;â&#x2020;&#x2019;â&#x2C6;&#x17E;
đ?&#x153;&#x2020;đ?&#x2018;&#x201C; (đ?&#x2018;&#x;) is non-negative and continuous for đ?&#x2018;&#x; â&#x2030;Ľ đ?&#x2018;&#x;0 , say đ?&#x153;&#x2020;đ?&#x2018;&#x201C; (đ?&#x2018;&#x;) is differentiable for đ?&#x2018;&#x; â&#x2030;Ľ đ?&#x2018;&#x;0 except possibly at isolated points at which đ?&#x153;&#x2020;đ?&#x2018;&#x201C;â&#x20AC;˛ (đ?&#x2018;&#x; + 0) and đ?&#x153;&#x2020;đ?&#x2018;&#x201C;â&#x20AC;˛ (đ?&#x2018;&#x; â&#x2C6;&#x2019; 0) exists, www.ijmer.com
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