A remark on the special classes of analytic functions

83

J. Korean Math. Soc.

Vo\. 19, No. 2, 1983

A REMARK ON THE SPECIAL CLASSES OF ANALYTIC FUNCTIONS

By ^{SHlGEYOSHI OWA}

1. Introduction

Let Q)(k) denote the class of functions

=

(1) fez) =z+

I:

allz"

12",;;2

analytic in the unit disk U-= {Jz

I

<I} and satisfying

I

^{f'(z) - ]}

I

<k

f'(z)+l ^(zEU)

for some k(O<k~l).

For this class Q)(k), K. S. Padmanabhan [6J and S. Owa [4J showed some results, respectively.

Let gJ(a, p) denote the class of functions (1) analytic and univalent in the unit disk 11 for which

I f'(~;t~;!2a) ^I

^<p,

where O~a'<l and O<p~1.

In 1976, V. P. Gupta and P. K. lain [2J gave a necessary and sufficient condition and some distortion theorems for a function

=

fez) =z-

I:

a"z"

n""-2

in the class gJ (a, f3).

Next, let gJ(a,

p,

r) denote the class of functions (1) analytic and univ- alent in the unit disk 11 for which

12r{f'(z)-C~f-~~?f'(z)

^-I}

^{I <fJ,}

where O~a<l, O<j3~l and _O<T~1.

In 1979, V. P. Gupta and1. Ahmad

[IJ

showed a necessary and sufficient condition and some distortion theorems for a function

=

fez) =z-

I:

a"z"

n==2

in the class (/jJ(a,

p,

r).

Received June 29. 1982.

Shigeyoshi Owa

Finally, let dCa,

(j)

denote the class of functions

(1)

analytic in the unit disk

U

for which

I z;~;) 11 _<~ I aj{:~z) ⁺ 1',

where

O;;;;;a<l

and

0<~;;;;;1.

For this class dCa,

(j),

T. V. Lakshminarasimhan [3J and S. Owa [5J gave some results.

2. Some theorems

THEOREM

1.

Let a function

00

fCz) =z+

1:;

anzn

n=2

be analytic in the unit disk U and

00

1:;

n(l +k)

I

an

I

;;;;;2k,

n=2 .

where O<k~l. Then the function fez) is in the class Q)(k).

Proof.

By using the hypothesis of the theorem, we have

II'(z) -ll-klf'Cz)

+11

⁼ IE2nanzn-II-k1

2+

E2nanzn-l/

<~2nranl-k {2- E2

^{nl anl}=}

~2n(1

+k) lanl-2k

;;;;;0.

Hence, by the maximum modulus theorem, the function

fez)

is in the class

Q)(k).

Furthermore, the function

fez) =z:'-

nCi~k)

^zn

is an extremal function for the theorem.

THEOREM

2.

Let-a function

00

fez) =z+

L;

anzn

n=2

be analytic and univalent in the unit disk U and

00

1:;

n(l

+m

lanl ;;;;;2fi(1-a),

n=2 -

where

O;;;;;a<l

and 0<~;;;;;1. Then the function fez) belongs to the class

5J

(a,fi).

Proof.

By using the condition of the theorem, we have

If'(z) -ll-~If'(z)+

Cl-2a) I

=1 ~ nanzn-ll-~ ^!2(1-a) + 13

^nanznc-l1_-

n-2 n=2

1

A remark on the special classes of analytic functions

00

<I;n(l

+/3)

lanl-2,8(1-a)~O.

n;;;;2

85

Therefore, by the maximum modulus theorem, the function

fez)

belongs to the class

:p(a,

/3). Moreover, the function

fez) =z- 2,8(1-a) zn n(l +/3)

is an extremal function with respect to the theorem.

THEOREM

3.

Let a function

00

fez) =z+I;anzn

1':1=:2

be analytic and univalent in the unit disk 11 and

00

I;n(l+2,8r-

/3)

IanI~2,8r(1-a),

12;;;;;2

where O~a<l, 0<,8~1 and O<r~l. Then the function f(z) is in the class :p(a,,8,r)·

Proof.

By the hypothesis of the theorem, we have

1f'(z)-11-,812rU'(z)-a} -

U'(z)-lll

=

^I

~2

^{nanzn-I!-,812rO-a)}

-.~

n(l-2r)anzn-^1I

00

<I; n(l +2;3r-;3)

I

an1-2;3r(l-a)

n;::2

~O.

Hence, we have the theorem with the aid of the maximum modulus theorem.

Furthermore, the function

f( ) 2,8r(1-a) n

z =z- n(l +2,8r-/3) z

is an extremal function with respect to the theorem.

THEOREM

4.

Let a function

00

fez) =z+

1::

anzn

n=::2

be analytic in the unit disk 11 and

00

I;{(n-l)+;3(l+an)}

lanl

~;3(l+a),

n=2

where O~a<l and 0<,8;;;;'1. Then the function fez) belongs to the class d (a,;3).

Proof.

By using the hypothesis of the theorem, we have I

zf' (z) - fez) 1-;31az!, (z)

+

^{fez) I}

=!f/n-l)anzn!-;3j (l+a)z+ntC1+an)anzn!

86 , Shigeyoshi Owa

~ ^Izl [f2{Cn-l)+,sCl+an)} lanl- Cl+a),s]

~O.

Hence, this completes the proof of the theorem with the aid of the ma- ximum modulus theorem. Moreover, the function

. ,sn

^+a)

^.

^zn

fCz) =zCn-l)

+,sCl +

an)

is an extremal function for the theorem.

References

1. V. P. Gupta and I. Ahmad, Certain classes of functions univalent in the unit disc II, Bull. Inst. Math. Acad. Sinica 7 (1979), 7-13.

2. V.P. Gupta and P.K. Jain, Certain classes of univalent functions with negative coefficients II, Bull. Austral. Math. Soc. 15 (1976), 467-473.

3. T. V. Lakshminarasimhan, On subclasses of functions starlike in the unit disc, ].

Indian Math. Soc. 41 (1977), 233-243.

4. S. Owa, On applications ofihe fractional calculus, Math. Japonica, 25 (1980), 195-206.

5. S. Owa, On the subclasses of univalent functions, Math. Japonica, 28(1983),97-108 6. K. S. Padmanabhan, A certain class of functions whose derivative have a positive

real part in the unit disk, Ann. Polon. Math. 23(1970), 73-81.

Department of Mathematics, Kinki University

Osaka, Japan