Some Families of Multivalently Analytic Functions with Negative Coefficients

Ž .JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 214, 674]690 1997ARTICLE NO. AY975615

Some Families of Multivalently Analytic Functionswith Negative Coefficients

Ming-Po Chen*

Institute of Mathematics, Academia Sinica, Taipei 11529, Taiwan, Republic of China

Huseyin Irmak†¨

Department of Mathematics, Faculty of Science and Letters, Kirikkale Uni ersity,TR-71450 Yahsihan-Kirikkale, Turkey

and

H. M. Srivastava‡

Department of Mathematics and Statistics, Uni ersity of Victoria,Victoria, British Columbia V8W 3P4, Canada

Submitted by George Leitmann

Received March 25, 1997

The authors introduce and study three novel subclasses of analytic and p-valentfunctions with negative coefficients. In addition to finding a necessary and suffi-cient condition for a function to belong to each of these subclasses, a number ofother potentially useful properties and characteristics of functions in these sub-classes are obtained. Finally, several applications involving an integral operator andcertain fractional calculus operators are also considered. Q 1997 Academic Press

* E-mail address: [email protected].† E-mail address: [email protected].‡ E-mail address: [email protected].

674

0022-247Xr97 $25.00Copyright Q 1997 by Academic PressAll rights of reproduction in any form reserved.

MULTIVALENTLY ANALYTIC FUNCTIONS 675

1. INTRODUCTION AND DEFINITIONS

Ž . Ž .Let TT n, p denote the class of functions f z of the form

`p k � 4f z s z y a z a G 0; n , p g N [ 1, 2, 3, . . . , 1.1Ž . Ž .Ž .Ý k k

ksnqp

which are analytic and p-valent in the open unit disk

< <� 4UU [ z : z g C and z - 1 .

Ž . Ž .A function f z g TT n, p is said to be p- alently starlike of order a in UU ifit satisfies the inequality

zf 9 zŽ .R ) a z g UU ; 0 F a - p; p g N . 1.2Ž . Ž .½ 5f zŽ .

Ž . Ž .On the other hand, a function f z g TT n, p is said to be p- alentlyconëx of order a in UU if it satisfies the inequality

zf 0 zŽ .R 1 q ) a z g UU ; 0 F a - p; p g N . 1.3Ž . Ž .½ 5f 9 zŽ .

Ž . Ž .Furthermore, a function f z g TT n, p is said to be p-valently close-to-conëx of order a in UU if it satisfies the inequality

R z1yp f 9 z ) a z g UU ; 0 F a - p; p g N . 1.4� 4Ž . Ž . Ž .

Ž .Motivated by these familiar subclasses of the class TT n, p of p-valentlyanalytic functions with negative coefficients, we aim here at presenting arather systematic investigation of the various important properties andcharacteristics of the following three novel families

SS p , a , l , KK p , a , l , and CC p , a , l ,Ž . Ž . Ž .n n n

Ž . Ž .where SS p, a , l denotes the subclass of TT n, p consisting of functionsnwhich also satisfy the inequality

zf 9 z y pf zŽ . Ž .- a

1 y l f z q l zf 9 zŽ . Ž . Ž .z g UU ; 0 - a F p; 0 F l F 1; p g N ; 1.5Ž . Ž .

CHEN, IRMAK, AND SRIVASTAVA676

Ž . Ž .KK p, a , l denotes the subclass of TT n, p consisting of functions whichnalso satisfy the inequality

zf 0 z y p y 1 f 9 zŽ . Ž . Ž .- a

l f 9 z q 1 y l zf 0 zŽ . Ž . Ž .z g UU ; 0 - a F p; 0 - l F 1; p g N ; 1.6Ž . Ž .

Ž . Ž .and CC p, a , l denotes the subclass of TT n, p consisting of functionsnwhich also satisfy the inequality

1ypR z f 9 z q l zf 0 z ) aŽ . Ž .� 4z g UU ; 0 F a - p l p y l q 1 ; l G 0; p g N . 1.7Ž . Ž .Ž .

Ž . Ž . Ž .By comparing the inequalities 1.5 , 1.6 , and 1.7 with the inequalitiesŽ . Ž . Ž .1.2 , 1.3 , and 1.4 , respectively, the special classes

SS p , a , 0 , KK p , a , 1 , and CC p , a , 0Ž . Ž . Ž .n n n

can easily be identified with the aforementioned classes of p-valentlystarlike functions of order p y a in UU, p-valently convex functions oforder p y a in UU, and p-valently close-to-convex functions of order a in

Ž w x.UU, respectively cf., e.g., Duren 4 .Ž .Numerous other interesting subclasses of the class TT n, p where inves-

Ž . w x w xtigated recently by for example Altintas et al. 1 , Chen and Aouf 3 ,w x w x Ž w xIrmak 5 , and Srivastava and Aouf 10 see also Srivastava and Owa 12

.for many earlier works on this subject .In our present investigation we shall make use of the familiar integral

Ž w x.operator J defined by cf. 2, 6, 7c, p

zc q pcy1J f z [ t f t dt f g TT n , p ; c ) yp; p g NŽ . Ž . Ž .Ž .Ž . Hc , p cz 0

1.8Ž .

as well as the fractional calculus operator D m for which it is well knownzŽ w x.that see, for details, 8, 9, 11

G r q 1Ž .m p rym� 4D z s z r ) y1; m g R 1.9Ž . Ž .z G r y m q 1Ž .

in terms of Gamma functions.


Ž .2. BASIC PROPERTIES OF THE CLASS SS p, a , ln

Ž . Ž .A necessary and sufficient condition for a function f z g TT n, p to beŽ .in the class SS p, a , l is given byn

Ž . Ž . Ž .THEOREM 1. Let a function f z defined by 1.1 be in the class TT n, p .Ž . Ž .Then the function f z belongs to the class SS p, a , l if and only ifn

`

k y p q a kl y l q 1 a F a pl y l q 1� 4Ž . Ž .Ý kksnqp

0 - a F p; 0 F l F 1; n , p g N . 2.1Ž . Ž .

Ž .The result is sharp for the function f z gi en by

a pl y l q 1Ž .p nqpf z s z y z n , p g N . 2.2Ž . Ž . Ž .

n q a 1 q l n q p y 1Ž .

Ž .Proof. Assuming that the inequality 2.1 holds true, if we let z g UU,Ž . Ž .we find from 1.1 and 2.1 that

zf 9 z y pf z y a 1 y l f z q l zf 9 zŽ . Ž . Ž . Ž . Ž .`

kyps y k y p a zŽ .Ý kksnqp

`kypy a pl y l q 1 y kl y l q 1 a zŽ . Ž .Ý k

ksnqp

`

F k y p q a kl y l q 1 a y a pl y l q 1� 4Ž . Ž .Ý kksnqp

F 0 0 - a F p; 0 F l F 1; n , p g N . 2.3Ž . Ž .

Ž .Hence, by the maximum modulus theorem, the function f z defined byŽ . Ž .1.1 belongs to the class SS p, a , l .n


Ž . Ž .In order to prove the converse, we suppose that f z g SS p, a , l , thatnis, that

zf 9 z y pf zŽ . Ž .1 y l f z q l zf 9 zŽ . Ž . Ž .

`kypk y p a zŽ .Ý k

ksnqps - a`kyppl y l q 1 y kl y l q 1 a zŽ . Ž .Ý k

ksnqp

z g UU ; 0 - a F p; 0 F l F 1; n , p g N . 2.4Ž . Ž .

< Ž . < < <Since R z F z for any z, choosing z to be real and letting z ª 1 yŽ .through real values, 2.4 yields

` `

k y p a F a pl y l q 1 y kl y l q 1 a , 2.5Ž . Ž . Ž . Ž .Ý Ýk k½ 5ksnqp ksnqp

Ž .which leads us immediately to the desired inequality 2.1 .Ž . Ž .Finally, by observing that the function f z given by 2.2 is indeed an

Ž .extremal function for the assertion 2.1 , we complete the proof ofTheorem 1.

The following inclusion property is an easy consequence of Theorem 1.

Ž .THEOREM 2. Let each of the functions f z defined byj

`p kf z s z y a z a G 0; j s 1, . . . , m; n , p g N 2.6Ž . Ž .Ž .Ýj k , j k , j

ksnqp

Ž . Ž .be in the class SS p, a , l . Then the function h z defined byn

m m

h z [ j f z j G 0 j s 1, . . . , m ; j s 1 2.7Ž . Ž . Ž . Ž .Ý Ýj j j jž /js1 js1

Ž .is also in the class SS p, a , l .n

Next we prove the following growth and distortion property for the classŽ .SS p, a , l .n


Ž . Ž .THEOREM 3. If f z g SS p, a , l , thenn

p! n q p ! a pl y l q 1Ž . Ž . n pym< < < <y z zž /p y m ! n q p y m ! n q a 1 q l n q p y 1� 4Ž . Ž . Ž .Žm.F f zŽ .

p!F ž p y m !Ž .

n q p ! a pl y l q 1Ž . Ž . n pym< < < <q z z/n q p y m ! n q a 1 q l n q p y 1� 4Ž . Ž .

� 4z g UU ; 0 - a F p; 0 F l F 1; n , p g N; m g N [ N j 0 ; p ) m .Ž .0

2.8Ž .

Ž . Ž .The result is sharp for the function f z gi en by 2.2 .

Ž . Ž .Proof. Under the hypothesis f z g SS p, a , l , we find from Theoremn1 that

`n q a 1 q l n q p y 1Ž .k! aÝ kn q p !Ž . ksnqp

`

F k y p q a kl y l q 1 a� 4Ž .Ý kksnqp

F a pl y l q 1 ,Ž .

which readily yields

` n q p ! a pl y l q 1Ž . Ž .k! a F n , p g N . 2.9Ž . Ž .Ý k n q a 1 q l n q p y 1Ž .ksnqp

Ž .Now, by differentiating both sides of 1.1 m times, we have

`p! k!Žm. pym kymf z s z y a zŽ . Ý kp y m ! k y m !Ž . Ž .ksnqp

n , p g N; m g N ; p ) m , 2.10Ž . Ž .0

Ž . Ž .and Theorem 3 would follow from 2.9 and 2.10 .

Finally, we determine the radii of p-valently starlikeness and p-valentlyŽ .convexity for functions in the class SS p, a , l , which are given byn


Ž . Ž . Ž .THEOREM 4. If f z g SS p, a , l , then f z is p- alently starlike ofnŽ . < < Ž .order b 0 F b - p in z - r and p- alently conëx of order b 0 F b - p1

< <in z - r , where2

Ž .1r kypp y b k y p q a kl y l q 1Ž . Ž .r s r p; a , b , l [ infŽ .1 1 ½ 5a k y b pl y l q 1k Ž . Ž .

k G n q p; n , p g N; 0 - a F p; 0 F b - p; 0 F l F 1 2.11Ž . Ž .

and

Ž .1r kypp p y b k y p q a kl y l q 1Ž . Ž .r s r p; a , b , l [ infŽ .2 2 ½ 5ka k y b pl y l q 1k Ž . Ž .

k G n q p; n , p g N; 0 - a F p; 0 F b - p; 0 F l F 1 . 2.12Ž . Ž .

Ž .Each of these results is sharp for the function f z gi en by

p pl y l q 1Ž .p kf z s z y z k G n q p; n , p g N .Ž . Ž .

k y p q a kl y l q 1Ž .2.13Ž .

Proof. It is sufficient to show that

zf 9 zŽ .< <y p F p y b z - r ; 0 F b - p; p g N 2.14Ž .Ž .1f zŽ .

and that

zf 0 zŽ .< <1 q y p F p y b z - r ; 0 F b - p; p g N 2.15Ž .Ž .2f 9 zŽ .

Ž . Ž . Ž .for a function f z g SS p, a , l , where r and r are defined by 2.11n 1 2Ž .and 2.12 , respectively. The details involved are fairly straightforward and

may be omitted.

Ž .3. BASIC PROPERTIES OF THE CLASSES KK p, a , lnŽ .AND CC p, a , ln

In this section we investigate the basic properties and characteristics ofŽ . Ž .functions belonging to the classes KK p, a , l and CC p, a , l . Since then n

derivations of these properties and characteristics would run parallel to


those of the corresponding results of Section 2, we merely summarize ourŽ . Ž .results for the classes KK p, a , l and CC p, a , l under Theorem 5 andn n

Theorem 6 below.

Ž . Ž . Ž .THEOREM 5. Let a function f z defined by 1.1 be in the class TT n, p .Ž . Ž .Then the function f z belongs to the class KK p, a , l if and only ifn

`

k k q p y 2 q a l q 1 y l k y 1 a� 4Ž . Ž .Ý kksnqp

F pa l q 1 y l p y 1Ž . Ž .0 - a F p; 0 - l F 1; n , p g N . 3.1Ž . Ž .

Ž . Ž .On the other hand, if f z g KK p, a , l , thenn

p!

p y m !Ž .�n q p y 1 ! pa l q 1 y l p y 1Ž . Ž . Ž . n pym< < < <y z zn q p y m ! n q 2 p y 1�Ž . Ž . 0

qa l q 1 y l n q p y 1 4Ž . Ž .Žm.F f zŽ .

p!F

p y m !Ž .�n q p y 1 ! pa l q 1 y l p y 1Ž . Ž . Ž . n pym< < < <q z zn q p y m ! n q 2 p y 1�Ž . Ž . 0

qa l q 1 y l n q p y 1 4Ž . Ž .

z g UU ; 0 - a F p; 0 - l F 1; n , p g N; m g N ; p ) m .Ž .0

3.2Ž .


f z s z pŽ .pa l q 1 y l p y 1Ž . Ž .

nqpy zn q p n q 2 p y 1 q a l q 1 y l n q p y 1� 4Ž . Ž . Ž . Ž .

n , p g N . 3.3Ž . Ž .


Ž . Ž . Ž .If each of the functions f z j s 1, . . . , m defined by 2.6 is in the classjŽ . Ž . Ž .KK p, a , l , then the function h z defined by 2.7 is also in the classnŽ .KK p, a , l .n

Ž . Ž . Ž .Furthermore, if f z g KK p, a , l , then f z is p- alently starlike of ordernŽ . < < Ž .b 0 F b - p in z - r and p- alently conëx of order b 0 F b - p in1

< <z - r , where2

r s r p; a , b , lŽ .1 1

Ž .1r kypp y b k k q p y 2 q a l q 1 y l k y 1� 4Ž . Ž . Ž .[ inf ½ 5pa k y b l q 1 y l p y 1k Ž . Ž . Ž .

3.4Ž .

and

r s r p; a , b , lŽ .2 2

Ž .1r kypp y b k q p y 2 q a l q 1 y l k y 1� 4Ž . Ž . Ž .[ inf ½ 5a k y b l q 1 y l p y 1k Ž . Ž . Ž .

k G n q p; n , p g N; 0 - a F p; 0 F b - p; 0 - l F 1 , 3.5Ž . Ž .Ž .the results being sharp for the functions f z gi en by

pa l q 1 y l p y 1Ž . Ž .p kf z s z y zŽ .

k k q p y 2 q a l q 1 y l k y 1� 4Ž . Ž .

k G n q p; n , p g N . 3.6Ž . Ž .Ž . Ž . Ž .THEOREM 6. Let a function f z defined by 1.1 be in the class TT n, p .

Ž . Ž .Then the function f z belongs to the class CC p, a , l if and only ifn`

k kl y l q 1 a F p pl y l q 1 y aŽ . Ž .Ý kksnqp

0 F a - p pl y l q 1 ; l G 0; n , p g N . 3.7Ž . Ž .Ž .Ž . Ž .On the other hand, if f z g CC p, a , l , thenn

p! n q p y 1 ! p pl y l q 1 y aŽ . Ž . n pym< < < <y z zž /p y m ! n q p y m ! 1 q l n q p y 1Ž . Ž . Ž .Žm.F f zŽ .

p! n q p y 1 ! p pl y l q 1 y aŽ . Ž . n pym< < < <F q z zž /p y m ! n q p y m ! 1 q l n q p y 1Ž . Ž . Ž .

z g UU ; 0 F a - p; l G 0; n , p g N; m g N ; p ) m . 3.8Ž . Ž .0



p pl y l q 1 y aŽ .p nqpf z s z y z n , p g N . 3.9Ž . Ž . Ž .

n q p 1 q l n q p y 1Ž . Ž .Ž . Ž . Ž .If each of the functions f z j s 1, . . . , m defined by 2.6 is in the classj

Ž . Ž . Ž .CC p, a , l , then the function h z defined by 2.7 is also in the classnŽ .CC p, a , l .n

Ž . Ž . Ž .Furthermore, if f z g CC p, a , l , then f z is p- alently starlike of ordernŽ . < < Ž .b 0 F b - p in z - r and p- alently conëx of order b 0 F b - p in1

< <z - r , where2

Ž .1r kypp y b k kl y l q 1Ž . Ž .r s r p; a , b , l [ infŽ .1 1 ½ 5k y b p pl y l q 1 y ak Ž . Ž .

3.10Ž .

and

Ž .1r kypp p y b kl y l q 1Ž . Ž .r s r p; a , b , l [ infŽ .2 2 ½ 5k y b p pl y l q 1 y ak Ž . Ž .

k G n q p; n , p g N; 0 F a - p pl y l q 1 ; 0 F b - p; l G 0 ,Ž .Ž .3.11Ž .

Ž .the results being sharp for the functions f z gi en by

p pl y l q 1 y aŽ .p kf z s z y z k G n q p; n , p g N . 3.12Ž . Ž . Ž .

k kl y l q 1Ž .

4. AN INCLUSION THEOREM INVOLVING THEMODIFIED HADAMARD PRODUCT

Ž . Ž . Ž .Let the function f z be defined by 1.1 . Also let the function g z bedefined by

`p kg z s z y b z b G 0; n , p g N . 4.1Ž . Ž . Ž .Ý k k

ksnqp

Ž .Then the modified Hadamard product or convolution of the functionsŽ . Ž .f z and g z is defined here by

`p kf ) g z [ z y a b z . 4.2Ž . Ž . Ž .Ý k k

ksnqp


By applying the Cauchy]Schwarz inequality, it is not difficult to prove theŽfollowing inclusion theorem involving the modified Hadamard product or

. Ž .convolution of functions belonging to the class CC p, a , l .n

Ž . Ž .THEOREM 7. If each of the functions f z and g z is in the classŽ . Ž .Ž . Ž .CC p, a , l , then f ) g z g CC p, g , l , wheren n

2p l p y l q 1 y aŽ .g F p pl y l q 1 y n , p g N .Ž . Ž .

n q p 1 q l n q p y 1Ž . Ž .4.3Ž .

Ž . Ž .The result is sharp for the functions f z and g z gi en precisely by the sameŽ .Eq. 3.9 .

5. PROPERTIES INVOLVING THE OPERATORSJ AND D m

c, p z

Ž . Ž . Ž .For a function f z g TT n, p defined by 1.1 , it is easily seen fromŽ . Ž .1.8 and 1.9 that

` c q pp kJ f z s z y a z c )yp; n , p g N 5.1Ž . Ž . Ž .Ž . Ýc , p kž /c q kksnqp

and

`G p q 1 G k q 1Ž . Ž .m pym kymD f z s z y a z� 4Ž . Ýz kG p y m q 1 G k y m q 1Ž . Ž .ksnqp

m g R; n , p g N . 5.2Ž . Ž .

Ž . Ž .Making use of 5.1 and 5.2 , we observe that

G p q 1Ž .m pymD J f z s zŽ .� 4Ž .z c , p G p y m q 1Ž .

` c q p G k q 1Ž . Ž .kymy a zÝ kc q k G k y m q 1Ž . Ž .ksnqp

m g R; c )yp; n , p g N 5.3Ž . Ž .


and

c q p G p q 1Ž . Ž .m pymJ D f z s z� 4Ž .Ž .c , p z c q p y m G p y m q 1Ž . Ž .

` c q p G k q 1Ž . Ž .kymy a zÝ kc q k y m G k y m q 1Ž . Ž .ksnqp

m g R; c )yp; n , p g N , 5.4Ž . Ž .

Ž .provided that no zeros appear in the denominators in 5.4 . Thus, ingeneral, the operators J and D m are non-commutati e.c, p z

In order to give growth and distortion properties of functions in theŽ . Ž . Ž .classes SS p, a , l , KK p, a , l , and CC p, a , l , involving the operatorsn n n

J and D m, we find it to be convenient to use the order of operationc, p zŽ . Ž . Ž .exhibited by 5.3 . If, for example, we suppose that f z g SS p, a , l , wen

Ž .find from the inequality 2.1 that

`

n q a 1 q l n q p y 1 a� 4Ž . Ý kksnqp

`

F k y p q a 1 q l k y 1 a� 4Ž .Ý kksnqp

F a 1 q l p y 1 ,Ž .

which readily yields

` a 1 q l p y 1Ž .a FÝ k n q a 1 q l n q p y 1Ž .ksnqp

0 - a F p; 0 F l F 1; n , p g N . 5.5Ž . Ž .If we set

c q p G k q 1Ž . Ž .Q k [ k G n q p; n , p g R , 5.6Ž . Ž . Ž .

c q k G k y m q 1Ž . Ž .

Ž .then it is easily seen that Q k is a decreasing function of k when m - 0,and hence

c q p G n q p q 1Ž . Ž .0 - Q k F Q n q p s m - 0 .Ž . Ž . Ž .

c q n q p G n q p y m q 1Ž . Ž .5.7Ž .


Ž . Ž . Ž .Now, by applying 5.5 and 5.7 in 5.3 , it is fairly straightforward toprove

Ž . Ž .THEOREM 8. If f z g SS p, a , l , thenn

¡ ¦G p q 1 a c q p 1 q l p y 1 G n q p q 1Ž . Ž . Ž . Ž . n pqm~ ¥< < < <y z z

G p q m q 1 c q n q p n q a 1 q l n q p y 1� 4Ž . Ž . Ž .¢ §?G n q p q m q 1Ž .

ymF D J f zŽ .� 4Ž .z c , p

¡G p q 1Ž .~F

G p q m q 1Ž .¢¦

a c q p 1 q l p y 1 G n q p q 1Ž . Ž . Ž . n pqm¥< < < <q z zc q n q p n q a 1 q l n q p y 1� 4Ž . Ž . §

?G n q p q m q 1Ž .

z g UU ; 0 - a F p; 0 F l F 1; m ) 0; c ) yp; n , p g N . 5.8Ž . Ž .

Ž .Next, for functions in the class KK p, a , l , we proven

Ž . Ž .THEOREM 9. If f z g KK p, a , l , thenn

w xG p q 1 pa c q p l q 1 y l p y 1 G n q pŽ . Ž . Ž .Ž . Ž . n pqm< < < <y z zw xG p q m q 1 c q n q p n q 2 p y 1 q a l q 1 y l p y 1� 4Ž . Ž . Ž . Ž .Ž .½ 5?G n q p q m q 1Ž .

ymF D J f zŽ .� 4Ž .z c , p

G p q 1Ž .F

G p q m q 1Ž .½w xpa c q p l q 1 y l p y 1 G n q pŽ . Ž .Ž . Ž . n pqm< < < <q z zw xc q n q p n q 2 p y 1 q a l q 1 y l p y 1� 4Ž . Ž . Ž .Ž . 5

?G n q p q m q 1Ž .

z g UU ; 0 - a F p; 0 - l F 1; m ) 0; c ) yp; n , p g N 5.9Ž . Ž .


and

w xG p q 1 pa c q p l q 1 y l p y 1 G n q pŽ . Ž . Ž .Ž . Ž . n pym< < < <y z zw xG p y m q 1 c q n q p n q 2 p y 1 q a l q 1 y l p y 1� 4Ž . Ž . Ž . Ž .Ž .½ 5?G n q p y m q 1Ž .

mF D J f zŽ .� 4Ž .z c , p

G p q 1Ž .F

G p y m q 1Ž .½w xpa c q p l q 1 y l p y 1 G n q pŽ . Ž .Ž . Ž . n pym< < < <q z zw xc y n q p n q 2 p y 1 q a l q 1 y l p y 1� 4Ž . Ž . Ž .Ž . 5

?G n q p y m q 1Ž .

z g UU ; 0 - a F p; 0 - l F 1; 0 F m - 1; c ) yp; n , p g N . 5.10Ž . Ž .

Ž . Ž .Each of these results is sharp for the function f z gi en by 3.3 .

Ž . Ž . Ž .Proof. Since f z g KK p, a , l , we can apply the inequality 3.1 tonconclude that

` pa l q 1 y l p y 1Ž . Ž .a F 5.11Ž .Ý k n q p n q 2 p y 1 q a l q 1 y l p y 1� 4Ž . Ž . Ž . Ž .ksnqp

and

` pa l q 1 y l p y 1Ž . Ž .ka FÝ k n q 2 p y 1 q a l q 1 y l p y 1Ž . Ž . Ž .ksnqp

0 - a F p; 0 - l F 1; n , p g N . 5.12Ž . Ž .Moreover, if we let

c q p G k Q kŽ . Ž . Ž .F k [ s k G n q p; n , p g R ,Ž . Ž .

c q k G k y m q 1 kŽ . Ž .5.13Ž .

Ž .then the function F k can easily be seen to be decreasing in k whenm - 1, and we have

c q p G n q pŽ . Ž .0 - F k F F n q p s m - 1 .Ž . Ž . Ž .

c q n q p G n q p y m q 1Ž . Ž .5.14Ž .


Ž . Ž . Ž .The assertion 5.9 follows when we apply 5.7 and 5.11 in conjunctionŽ . Ž .with 5.3 . The assertion 5.10 , on the other hand, is a consequence of

Ž . Ž . Ž .5.3 , 5.12 , and 5.14 .

Ž .Finally, for functions in the class CC p, a , l , we shall proven

Ž . Ž .THEOREM 10. If f z g CC p, a , l , thenn

G p q 1Ž .½ G p q m q 1Ž .

c q p p 1 q l p y 1 y a G n q p� 4Ž . Ž . Ž . n pqm< < < <y z z5c q n q p 1 q l n q p y 1 G n q p q m q 1Ž . Ž . Ž .ymF D J f zŽ .� 4Ž .z c , p

G p q 1Ž .F ½ G p q m q 1Ž .

c q p p 1 q l p y 1 y a G n q p� 4Ž . Ž . Ž . n pqm< < < <q z z5c q n q p 1 q l n q p y 1 G n q p q m q 1Ž . Ž . Ž .

z g UU ; 0 F a - p pl y l q 1 ; l G 0; m ) 0; c ) yp; n , p g NŽ .Ž .5.15Ž .

and

G p q 1Ž .½ G p y m q 1Ž .

c q p p 1 q l p y 1 y a G n q p� 4Ž . Ž . Ž . n pym< < < <y z z5c q n q p 1 q l n q p y 1 G n q p y m q 1Ž . Ž . Ž .mF D J f zŽ .� 4Ž .z c , p

G p q 1Ž .F ½ G p y m q 1Ž .

c q p p 1 q l p y 1 y a G n q p� 4Ž . Ž . Ž . n pym< < < <q z z5c q n q p 1 q l n q p y 1 G n q p y m q 1Ž . Ž . Ž .

z g UU ; 0 F a - p pl y l q 1 ; l G 0; 0 F m - 1; c ) yp; n , p g N .Ž .Ž .5.16Ž .


Ž . Ž .Each of these results is sharp for the function f z gi en by 3.9 .

Ž . Ž . Ž .Proof. For f z g CC p, a , l , the inequality 3.7 readily yieldsn

` p 1 q l p y 1 y aŽ .a F 5.17Ž .Ý k n q p 1 q l n q p y 1Ž . Ž .ksnqp

and

` p 1 q l p y 1 y aŽ .ka FÝ k 1 q l n q p y 1Ž .ksnqp

0 F a - p pl y l q 1 ; l G 0; n , p g N . 5.18Ž . Ž .Ž .

Ž . Ž . Ž .Now the assertion 5.15 is proven by applying 5.7 and 5.17 in conjunc-Ž . Ž . Ž . Ž .tion with 5.3 . And, if we apply 5.14 and 5.18 in conjunction with 5.3 ,

Ž .we shall arrive at the assertion 5.16 .

We conclude by remarking that each of the results of this section can beextended appropriately to hold true for much more general known opera-

m Ž .tors of fractional calculus than the operator D m g R which we havezŽ w x.considered here cf., e.g., 3, 10, 12 . The details involved are fairly

straightforward and may be left as an exercise for the interested reader.

ACKNOWLEDGMENTS

The present investigation was completed during the third-named author’s visits to theŽ .Institute of Mathematics Academia Sinica at Taipei, Tamkang University at Tamsui,

Soochow University at Taipei, National Tsing Hua University at Hsin-Chu, and Chung YuanChristian University at Chung-Li in December 1996 and January 1997. This work wassupported, in part, by the National Science Council of the Republic of China under GrantNSC-86-2115-M-001-001 and, in part, by the Natural Sciences and Engineering ResearchCouncil of Canada under Grant OGP0007353.

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Some Families of Multivalently Analytic Functions with Negative Coefficients