Izl · 2019. 8. 1. · Internat. J. Math. & Math. Sci. VOL. 13 NO. 3 (1990) 621-623 621...

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Internat. J. Math. & Math. Sci. VOL. 13 NO. 3 (1990) 621-623 621 AN APPLICATION OF MILLER AND MOCANU’S RESULT SHIGEYOSHI OWA Department of Mathematics Kinki University Higashi-Osaka 57 7 Ja pan ZHWOREN WU Department of Mathematics Tongji University Shanghai, People’s Republic of China (Received April 7, 1988 and in revised form October 15, 1988) ABSTRACT. The object of the present paper is to give an application of Miller and Mocanu’s result for a certain integral operator. KEY WORDS AND PHRASES. Integral operator, Miller and Mocanu’s result, set H complex valued function. 1980 AMS SUBJECT CLASSIFICATION CODE. 30C45. I. INTRODUCTION. Let A denote the class of functions of the form f(z) z + a z (I.I) n n=2 which are analytic in the unit disk U {z: Izl < I}. For a function f(z) belonging to the class A, we define the integral operator J by [I p. 126, Equation (2.1)] c J (f(z)) c + tc-lf(t)dt (c > -I). (1.2) c c z 0 The operator J when c N {1,2,3,...}, was introduced by Bernardi [2]. In c particular, the operator J1 was studied earlier by Libera [3] and Livingston [4]. Recently, Owa and Srlvastava [I] have proved a property of the operator J (c > -I). c With the above operator J we now introduce c DEFINITION. Let H be the set of complex-valued functions h(r,s,t); c h(r,s,t): ? C (C is the complex plane)

Transcript of Izl · 2019. 8. 1. · Internat. J. Math. & Math. Sci. VOL. 13 NO. 3 (1990) 621-623 621...

Page 1: Izl · 2019. 8. 1. · Internat. J. Math. & Math. Sci. VOL. 13 NO. 3 (1990) 621-623 621 ANAPPLICATIONOFMILLERANDMOCANU’SRESULT SHIGEYOSHIOWA Department of Mathematics Kinki University

Internat. J. Math. & Math. Sci.VOL. 13 NO. 3 (1990) 621-623

621

AN APPLICATION OF MILLER AND MOCANU’S RESULT

SHIGEYOSHI OWA

Department of MathematicsKinki UniversityHigashi-Osaka 57 7

Japan

ZHWOREN WUDepartment of Mathematics

Tongji UniversityShanghai, People’s Republic of China

(Received April 7, 1988 and in revised form October 15, 1988)

ABSTRACT. The object of the present paper is to give an application of Miller and

Mocanu’s result for a certain integral operator.

KEY WORDS AND PHRASES. Integral operator, Miller and Mocanu’s result, set H complex

valued function.

1980 AMS SUBJECT CLASSIFICATION CODE. 30C45.

I. INTRODUCTION.

Let A denote the class of functions of the form

f(z) z + a z (I.I)n

n=2

which are analytic in the unit disk U {z: Izl < I}. For a function f(z) belonging

to the class A, we define the integral operator J by [I p. 126, Equation (2.1)]c

J (f(z)) c + tc-lf(t)dt (c > -I). (1.2)c c

z 0

The operator J when c N {1,2,3,...}, was introduced by Bernardi [2]. Inc

particular, the operator J1 was studied earlier by Libera [3] and Livingston [4].

Recently, Owa and Srlvastava [I] have proved a property of the operator J (c > -I).c

With the above operator J we now introducec

DEFINITION. Let H be the set of complex-valued functions h(r,s,t);c

h(r,s,t): ? C (C is the complex plane)

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622 S. OWA AND Z. WU

such that

(i) h(r,s,t) is continuous in D(L C3;

(ii) (0,0,0) g D and [h(O,O,O) < I;

(ill) }h(e iO m+c iO iOc- e me + L > whenever

i 0 m+c i 0 i 0+e me L) e D with Re(e-iOL) ) m(m-l)= forc+l

real 0 and real m I.

2. AN APPLICATION OF MILLER AND MOCANU’S RESULT.

We begin with the statement of the following lemma due to Miller and Mocanu [5].

LEMMA. Let w(z) e A with w(z) 0 in U. If z0 roeis0 (0 < r

0 < I) and

z0w’ (z0) mw(z0)and z

0w" z0)

Re {I +} m,w’ zO)where m is real and ) I.

(2.2)

and

Applying the above lemma, we derive the following

THEOREM. Let h(r,s,t) H and let f(z) A satisfyc

(Jc(f(z)),f(z),zf’(z)) Dr--.. C3

{h(J (f(z)) f(z)zf’(z)){ <

for c > -1 and U. Then we have

IJ (f(z)){ < (z e U)c

where J (f(z)) is defined by (1.2).c

PROOF. Letting J (f(z)) w(z) for f(z) e A, we have w(z) E A andc

w(z) 0 (z e U). Since

Z(Jc(f(z)))’ (c + l)f(z) -CJc(f(z)),we have

and

c w’f(z) =w(z) +z (z)

zf’(z) zw’ (z) + z2’(z).

Suppose that there exists a point z0 r0el80 (0 < r

0 < I) such that

max

Then, using the above lemma, we have

i80 m+c i80J (f(z0)) e f(z0) e z0f’(z0) me 180 + L,

(2.3)

(2.4)

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APPLICATION OF MILLER AND MOCANU’S RESULT 623

where L z20w"(z0)/(c+l). Further, applying the lemma, we see that

that is,

zoW" z0) zw" z0Re{ w,z0 Re meI0-------- m- I,

Re(e-i L) re(m-l) (2 7)c+l

Therefore, the condition h(r,s,t) H implies thatc

i00 m+c i0 i00c c+

(2.)

This contradicts our condition (2.3). nsequently, we conclude that

w(z)[ Jc(f(z)) < for all z U. us we complete the proof of the assertion

of the theorem.

Taking c 0 in the theorem, we have the following

COROLLARY. Let h(r,s,t) H0, and let f(z) A satisfy

(F(z),zF’(z),z(zF’(z))’) g DC C3and

lh(F(z),zF’(z),z(zF’(z))’)l <

Then IF(z)l < (z g U), where

F(z) J0(f(z))-- dr.0

(z u). (2.9)

ACKNOWLEDCMENT.

The authors of the paper would llke to thank the referee for his comments.

REFERENCES

I. OWA, S. and SRIVASTAVA, H.M., Some applications of the generalized Liberaintegral operator, Proc. Japan. Acad. set. A Math. Scl. 62 (1986), 125-128.

2. BERNARDI, S.D., Convex and starlike univalent functions, Trans. Amer. Math. Soc.135 (1969), 429-446.

3. LIBERA, R.J. Some classes of regular univalent functions, Proc. Amer. Math.Soc.16 (196 5), 755-758.

4. LIVINGSTON, A.E., On the radius of unlvalence of certain analytic functions,Proc. Amer. Math. Soc. 17 (1966), 352-357.

5. MILLER, S.S. and MOCANU, P.T., Second order differential inequalities in thecomplex plane, J. Math. Anal. Appl. 65 (1978), 289-305.

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