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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

    Japan

    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)

  • 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 < r0 < I) and

    z0w’ (z0) mw(z0) and z

    0 w" 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 satisfy c

    (Jc(f(z)),f(z),zf’(z)) Dr--.. C 3

    {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 and

    c 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 < r0 < 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)

  • 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 that c

    i00 m+c i0 i00 c 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 C 3 and

    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 Libera integral 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 the complex plane, J. Math. Anal. Appl. 65 (1978), 289-305.

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