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A subordination result with Salagean-type certain analytic functions of complex order.

1 Introduction

Let A denote the class of functions of the form

(1.1) f(z) = z + ([infinity].summation over (k=2)] [a.sub.k][z.sup.k]

which are analytic in the open unit disk U = {z [member of] C : [absolute value of c] < 1}. Also, let C denote

the familiar class of functions f(z) [member of] A which are convex in U. Salagean [3] has introduced the following operator called the Salagean operator:

[D.sup.0] f(z) = f(z)

[D.sup.1] f(z) = Df(z) = zf'(z)

[D.sup.n]f(z) = D ([D.sup.n-1]f(z)), n [member of] [N.sub.0] = {0,1,2, ...}.

With the help of the Salagean operator [D.sup.n], we say that a function f{z) belonging to A is in the class [H.sub.n] (b, M) iff [D.sup.n]f(z)/z [not equal to] 0 in U, and

(1.2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

M > 1/2 and b [note equal to] 0; complex. The class [H.sub.n](b,M) was introduced by Aouf et.al. [1].

They showed that

f [member of] [H.sub.n](b,M) if and only if [D.sup.n] f [member of] [H.sub.0](b,M) = F(b,M),

the class F(b, M) of bounded starlike functions of complex order was introduced by Nasr and Aouf [2].

Aouf et.al. [1] proved that if the function f{z) defined by (1.1) and

(1.3) [[infinity].summation over (k=2)] {k - 1 + [absolute value of b(1 + m) + m(k - 1)]} [k.sup.n][absolute value of [a.sub.k]] [less than or equal to][absolute value of b(1 + m)]

hold then f(z) belongs to [H.sub.n](b,M), where m = 1 - 1/M (M > 1/2).

Let [[alpha].sub.j] (j = 1,2, ..., p) and [[beta].sub.j](j = 1,2, ..., q) be complex numbers with

[[beta].sub.j] [not equal to] 0, -1, -2, ...; j = 1,2, ..., q.

The generalized hypergeometric function [sub.p][F.sub.q] is defined by (cf.[4, p.33])

(1.4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

where [([mu]).sub.k] is the Pochhammer symbol defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We note that the [.sub.p][F.sub.q] series in (1.4) converges absolutely for [absolute value of z] < [infinity] if p < q + 1, and for z [member of] U if p = q + 1.

Let [K.sub.n] (b, M) denote the class of functions f{z) [member of] A whose coefficients satisfy the condition (1.3).

We note that

[K.sub.n](b,M) [subset or equal to] [H.sub.n](b,M).

We can show that:

Example

i) Let b [not equal to] 0; complex and m = 1 - 1/M (M > 1/2; M [not equal to] 1), then [f.sub.0](z) [member of] [K.sub.n](b, M), where

[D.sup.n][f.sub.0](z) = z(1 - mz) -b(1+m)/m (z [member of U)

which gives

[f.sub.o](z)= z[l + b(1 + m)z n+2 [f.sub.n+1](1, ..., 1, b(1+m)/m + 1; 2, ..., 2; mz)] (z [member of] U)

for real b with b [not equal to] 0.

ii) Let b [not equal to] 0; complex and m = 1 - 1/M (M > 1/2), the functions

[f.sub.1](z) = z [+ or -] (1 + m)[absolute value of b]/4(1 + [absolute value of m + b(1 + m])[z.sup.2]

and

[f.sub.2](z) = z [+ or -] (1 + m)[absolute value of b]/9(2 + [absolute value of 2m + b(1 + m])[z.sup.2]

[f.sub.2](z) = z [+ or -] (1 + m)[absolute value of b]/9(2 + [absolute value of 2m + b(1 + m)][z.sup.2]

are members in the class [K.sub.n] (b, M).

In this paper, we prove an interesting subordination result for the class [K.sub.n] (b,M). In our proposed investigation of functions in the class [K.sub.n] (b,M), we need the following definitions and lemma.

Definition 1. Given two functions f, g [member of] A where f(z) is given by (1.1) and g(z) is defined by

g(z) = z [[infinity].summation over (k=2)] [b.sub.k] [z.sup.k]

The Hadamard product f * g is defined by

(f * g)(z) = z + [[infinity].summation over (k=2)] [a.sub.k] [b.sub.k] [z.sup.k] (z [member of] U).

Definition 2. (Subordination Principle) For two functions f and g analytic in U, we say that the function f{z) is subordinate to g(z) in U and write f(z) [??] g(z), z [member of] U, if there exists a Schwarz function w(z), analytic in U with w(0) = 0 and [absolute value of w{z)] < 1, such that f(z) = g(w(z)), z [member of] U. In particular, if the function g(z) is univalent in U, the above subordination is equivalent to f(0) = g(0) and f(U) [subset or equal to] g (U).

Definition 3. (Subordinating Factor Sequence) A sequence [{[b.sub.k]}.sup.[infinity.sub.k=1] of complex numbers is said to be a Subordinating Factor Sequence if for the function f(z) of the form (1.1) is analytic, univalent and convex in U, we have the subordination given by

(1.5) [[infinity].summation over (k=1)] [a.sub.k] [b.sub.k] [z.sub.k] < f(z) (z [member of] U; [a.sub.1] = 1).

Lemma. The sequence {[b.sub.k]}.sup.[infinity].sub.k=1] is Subordinating factor sequence iff

(1.6) Re { 1 + 2 [[infinity].summation over (k = 1)] [b.sub.k] [z.sup.k]} > 0 (z [member of] U).

The above lemma is due to Wilf [5].

2 Main Theorem

Theorem. Let m = 1 - 1/M (M > 1/2). Also, let b [not equal to] 0; complex with Re(b) > -m/2(1+m) when m > 0 and Re(b) < 2 (-m)/2(1+m) when m < 0. If f(z) [member of] [K.sub.n](b,M) then

(2.1) [(1 + [absolute value of b (1 + m) + m]) [2.sup.n-1]/ [(1 + [absolute value of b (1 + m) + m] [2.sup.n] + [absolute value of b (1 + m)] (f x g) (z) < g (z)

(z [member of] U; n [member of] [N.sub.0]; g(z) [member of] C)

and

(2.2) Ref(z) > - 1 - (1 + m) [absolute value of b]/ (1 + [absolute value of b (1 + m) + m]) [2.sup.n].

The constant (1 + [absolute value of b (1 + m) + m]) [2.sup.n-1]/[(1 + [absolute value of b (1 + m] + m] [2.sup.n] + [absolute value of [b.sub.(1 + m)] is the best estimate.

Proof. Let/(z) [member of] [k.sub.n] (b, M) and g(z) = z + [[SIGMA].sup.[infinity].sub.k=2] [c.sub.k][z.sup.k] [member of] C. Then

(1 +[absolute value of b (1 + m) + m]) [2.sup.n-1]/ [(1 + [absolute value of b (1 + m) + m]) [2.sup.n] + [absolute value of b (1 + m)] (f x g) (z)

(2.3) (1 +[absolute value of b (1 + m) + m]) [2.sup.n-1]/ [(1 + [absolute value of b (1 + m) + m]) [2.sup.n] + [absolute value of b (1 + m)] (z + [[infinity].summation over (k = 2)] ([a.sub.k] [c.sub.k] [z.sup.k]).

Thus, by definition 3, (2.1) will hold true if

(2.4) {(1 + [absolute value of b (1 + m) + m]) [2.sup.n-1]/ (1 + [absolute value of b (1 + m) + m]) [2.sup.n-1] + [absolute value of b (1 + m)] [a.sub.b]}.sup.[infinity].sub.k=1]

Thus, by definition 3, (2.1) will hold true if

is a subordinating factor sequence with [a.sub.1] = 1. In view of Lemma, this is equivalent to

(2.5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Now because {k - 1 + [absolute value of b (1 + m)] + m (k - 1)]} [k.sup.n] (n [member of] [N.sup.0]; k [greater than or equal to] 2) is increasing function of ft, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence, (2.5) holds true in U and also the subordination result (2.1) asserted by Theorem 1. The inequality (2.2) follows by taking g(z) = [z/1-z] = [[SIGMA].sup.[infinity].sub.k=1] [z.sup.k] in (2.1).

Now, consider the function

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which is a member of the class [K.sub.n] (b, M). Then by using (2.1), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

It is easily verified that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Then the constant [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] cannot be replaced by a larger one, which completes the proof of Theorem.

References

[1] M.K. Aouf, H.E. Darwish and A. A. Attiya, On a class of certain analytic functions of complex order, Indian J.Pure App.Math, 32(2001), no.10,1443-1452.

[2] M.A. Nasr and M.K. Aouf, Bounded starlike functions of complex order, Proc. Indian Acad. Sci. Math. Sci.92(1983), no.2, 97-102.

[3] G.S. Salagean, Subclasses of univalent functions, Lecture Notes in Mathematics, 1013-Springer Verlag, Berlin Heidelberg and Newyork, (1983), 362-372.

[4] H.M.Srivastava and B.R.K. Kashyap,Special Functions in Queuing Theory and Related Stochastic Processes, Academic Press, New York and London, 1982.

[5] H.S. Wilf, Subordinating factor sequences for convex maps of the unit circle, Proc. Amer. Math. Soc. 12(1961), 689-693.

Received by the editors November 2009.

Communicated by F. Brackx.

2000 Mathematics Subject Classification : Primary 30C45; Secondary 33C20.

Department of Mathematics,

Faculty of Science, University of Dicle,

21280, Diyarbakir, TURKEY

email:ozlemg@dicle.edu.tr

Department of Mathematics,

Faculty of Science, University of Mansoura,

Mansoura 35516, EGYPT

Current address: Department of Mathematics

College of Science, King Khalid University

Abha, P.O. Box 9004, Saudi Arabia

email:aattiy@mans.edu.eg
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Author:Guney, H. Ozlem; Attiya, A.A.
Publication:Bulletin of the Belgian Mathematical Society - Simon Stevin
Article Type:Report
Date:May 1, 2011
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