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Extremal problems for functions of positive real part and applications

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Anh, Vo Van,1949- (1978) Extremal problems for functions of positive real part and applications. PhD thesis, University of Tasmania.

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Abstract

Let B be the class of functions w(z) regular in |z|< I and
satisfying w(0) = 0 , lw(z)I < 1 in Izl < I. We denote by P(A, B),
-I ≤ B < A ≤ I, the class of functions p(z) = l+p l z+... regular in lz1 < I
and such that p(z)=[1+Aw(z)]/[1+Bw(z)]for some w(z) ε B. This thesis is
concerned with establishing bounds on z1=r<1 for functionals of the form
Re{ap(z) + Up l (z)/p(z)} , a,β real ,

where p(z) varies in P(A, B) or one of the following subclasses:
Pk (A, B) = {p(z) = 1 +Pkzk+P2kz2k+— ε P(A, B), k = 1,2,3,...}
Pb (A, B) = {P(z) ε P(A2 B) ; P ' (0) = b(A-B) , 0 ≤ b ≤ 1} ,
P [a,b] = {P(z) ε P = P(1,-1) p(a)=b , 0 < a < I , b > 0}
The bounds obtained are used to derive the distortion theorems,
the covering theorems and the radii of convexity for the classes of regular
or meromorphic starlike functions associated with P(A, B) or the
above-mentioned subclasses.
Furthermore, we obtain bounds for the functional Re{p(z)a-azp 1 (z)/p(z)},
0 < a ≤ 1 , p(z) E p , and establish the above theorems for the class of
meromorphic strongly starlike functions of order a.
The problem of minimising the functional Re{zp' (z)/p(z)} over P(A, B)
is also examined for the case in which we may have A ≥ I . This situation
arises from the investigation of the starlikeness of functions f(z)
normalised, regular in Iz| < I and satisfying |f(z)/[Af(z)+(1-A)g(z)]-y|<y ,
y ≥ 1, 0 ≤ A < 1 , where g(z) belongs to, for example, the class S*a
of starlike functions of order a .
Finally we investigate the β -convexity of certain subclasses of
starlike functions. In particular, the radii of b-convexity, a real,
for the class S*a, 0 ≤ a ≤ 1/2 , and the class S*a[a] = {f(z) =z+a 2 z2
+...;lzf'(z)/f(z)-1|/|zfl(z)/f(z)+1|<a, 0<a<1, zεΔ},
are completely determined .

Several results of Chapter 1 are to appear in J. Math. Anal Appl.
( see [92]). Most of the material of Chapter 6 has been published in
Pacific J. Math. ( see [4]). Results of Chapter 7 have been submitted
for publication ( see [5]).

Item Type: Thesis (PhD)
Keywords: Analytic functions, Star-like functions
Copyright Holders: The Author
Copyright Information:

Copyright 1978 the Author - The University is continuing to endeavour to trace the copyright
owner(s) and in the meantime this item has been reproduced here in good faith. We
would be pleased to hear from the copyright owner(s).

Additional Information:

Thesis (Ph.D.)--University of Tasmania, 1978. Bibliography: l. 185-192

Date Deposited: 25 Nov 2014 00:38
Last Modified: 08 Jun 2016 02:59
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