CANA.pdf


CERTAIN BOUNDS FOR A SUBCLASSES OF ANALYTIC 

FUNCTIONS OF RECIPROCAL ORDER 

K. DHANALAKSHMI1, D. KAVITHA2•*, AND K.KANOHANA3

ABSTRACT. In this paper, we introduce certain class of bi univalent functionsrelated to shell like curves connected with Fibonacci numbers. Also we determineinitial Taylor-Maclaurin coefficient inequalities and Fekete Szego problem for thebelonging class. Mathematics Subject Classification: 30045,30050 Keywords: analytic functions, bi-univalent, shell-like curve, Fibonacci numbersMathematics Subject Olassification:Primary 30045 ; Secondary 30050

1. INTRODUCTION AND PRELIMINARIES

Let A denote the family of normalized analytic functions l of the form
00 

(z E 1U)
k=2 

(1.1)
in the open disc 1U = { z : z E C : I z I < 1}. Further, let S denote the class of
functions in A which are also univalent in 1U.

The well-known Koebe one-quarter theorem [2] ensures that the image of 1U
under every univalent function l E A contains a disk of radius 1 / 4. Hence every
univalent function l has an inverse 1- 1 satisfying 1- 1(f(z)) = z, (z E 1U) and

1-
1(f(w)) = w, (lwl < ro(f),ro(f) 2 1/4)

where,
g(w) = 1-

1(w) = w - a2w2 + (2a� - a3)w3 - (5a� - 5a2a3 + a4)w4 +... (1.2)
A function l EA is said to be bi-univalent in 1U if both land 1- 1 are univalent

in 1U. Let :E denote the class of bi-univalent functions in 1U given by (1.1). For
example, functions in the class :E are given below [13]:

1 � z' -log(l - z), �log(��;).

In 1967, Lewin [8] introduced the class :E of bi-univalent functions and shown that
la2 1 < 1.51. In 1969, Netanyahu [10] showed that max/EEla2 1 = 4/3 and Suffridge
[14] have given an example of l E :E for which la2 1 = 4/3. Later, in 1980, Brannan

Communications on Applied Nonlinear Analysis 
ISSN: 1074-133X 
Vol 32 No. 10s (2025) 

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Article History: Received: 12-01-2025, Revised: 15-02-2025, Accepted: 01-03-2025



Communications on Applied Nonlinear Analysis 
ISSN: 1074-133X 
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Communications on Applied Nonlinear Analysis 
ISSN: 1074-133X 
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ISSN: 1074-133X 
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Communications on Applied Nonlinear Analysis 
ISSN: 1074-133X 
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ISSN: 1074-133X 
Vol 32 No. 10s (2025) 

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Communications on Applied Nonlinear Analysis 
ISSN: 1074-133X 
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Communications on Applied Nonlinear Analysis 
ISSN: 1074-133X 
Vol 32 No. 10s (2025) 

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Communications on Applied Nonlinear Analysis 
ISSN: 1074-133X 
Vol 32 No. 10s (2025) 

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