VI.—The Maximum Term of an Entire Series

1953 ◽  
Vol 64 (1) ◽  
pp. 80-89
Author(s):  
S. M. Shah ◽  
S. K. Singh
Keyword(s):  
Entire Function ◽  
Infinite Order ◽  
Maximum Modulus ◽  
Zero Order ◽  
Maximum Term ◽  
Entire Series

SynopsisThe relation between the maximum term and the maximum modulus of an entire function is exhibited by means of general theorems and specific examples. Functions of zero order and of infinite order are mainly considered.

The Maximum Term and the Rank of an Entire Function

1969 ◽  
Vol 21 ◽  
pp. 257-261
Author(s):  
V. Sreenivasulu
Keyword(s):  
Entire Function ◽  
Maximum Modulus ◽  
Maximum Term ◽  
Proximate Order ◽  
Image Position

1. For an entire function , let M(r, f), μ(r, f), and v(r, f) denote the maximum modulus, the maximum term, and the rank for |z\ = r, respectively. Also, letand λ(r) the lower proximate order relative to log M(r, f). For the properties of the lower proximate order, we refer the reader to the paper by Shah (1).2. We prove the following theorems.THEOREM 1. For an entire functionwhere μ(r, f1) and M(r, f1) correspond to fl(z), the derivative of f(z), provided (n + l)Rn > nRn+1, when L(f) > 1.

On the Properties of an Entire Function of Two Complex Variables

1968 ◽  
Vol 20 ◽  
pp. 51-57
Author(s):  
Arun Kumar Agarwal
Keyword(s):  
Entire Function ◽  
Double Series ◽  
Complex Variables ◽  
Maximum Term ◽  
Image Position

1. Letbe an entire function of two complex variables z1 and z2, holomorphic in the closed polydisk . LetFollowing S. K. Bose (1, pp. 214-215), μ(r1, r2; ƒ ) denotes the maximum term in the double series (1.1) for given values of r1 and r2 and v1{m2; r1, r2) or v1(r1, r2), r2 fixed, v2(m1, r1, r2) or v2(r1, r2), r1 fixed and v(r1r2) denote the ranks of the maximum term of the double series (1.1).

On the Maximum Modulus and the Mean Modulus of an Entire Function

1969 ◽  
Vol 12 (6) ◽  
pp. 869-872 ◽  
Author(s):  
A.R. Reddy
Keyword(s):  
Entire Function ◽  
Maximum Modulus ◽  
The Mean ◽  
Image Position

Let be an entire function, but not a polynomial. As usual let,1

On the question of whether f″+ e−zf′ + B(z)f = 0 can admit a solution f ≢ 0 of finite order

1986 ◽  
Vol 102 (1-2) ◽  
pp. 9-17 ◽  
Author(s):  
Gary G. Gundersen
Keyword(s):  
Entire Function ◽  
Finite Order ◽  
Infinite Order ◽  
Independent Interest ◽  
Partial Result ◽  
Transcendental Entire Function ◽  
Degree Polynomial ◽  
Odd Degree ◽  
Polynomial Of Odd Degree

SynopsisWe show that if B(z) is either (i) a transcendental entire function with order (B)≠1, or (ii) a polynomial of odd degree, then every solution f≠0 to the equation f″ + e−zf′ + B(z)f = 0 has infinite order. We obtain a partial result in the case when B(z) is an even degree polynomial. Our method of proof and lemmas for case (i) of the above result have independent interest.

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