MAT~NAT~CS AND

C©N PUTATE©N ELSEVIER Applied Mathematics and Computation 91 (1998) 285-296

Some operators of fractional calculus and their applications involving a novel class

of analytic functions Ming-Po Chen a,1, H.M. Srivastava b,,, Ching-Shu Yu c,2

Institute of Mathematics, Academia Sinica, Taipei 11529, Taiwan, ROC b Department of Mathematics and Statistics, University of Victoria, Victoria, BC, Canada V8W3P4 c Department of Mathematics, Chung Yuan Christian University, Chung-Li 32033, Taiwan, ROC

Abstract

Making use of certain operators of fractional calculus (that is, fractional integral and fractional derivative), generalizations of various growth-and-distortion type results in terms of a novel class of analytic functions are presented. These general results are shown to stern naturally from some recent conjectures and theorems in Geometric Func- tion Theory. © 1998 Elsevier Science Inc. All rights reserved.

AMS classification: 26A33; 30C45; 33C20

Keywords: Fractional calculus; Growth-and-distortion type theorems; Analytic functions; Geo- metric function theory; Univalent functions; Koebe function; L6wner's result; Removable singularities; Gauss hypergeometric function; Generalized hypergeometric function; Euler's transformation; Chu-Vandermonde summation theorem

I. Introduction and preliminaries

Let s¢ d e n o t e the class o f f u n c t i o n s f ( z ) normalized by

f ( 0 ) = f ' ( 0 ) - 1 = 0,

so tha t

* Corresponding author. E-mail: hmsri@uvvm.uvic.ca. t Deceased on 9 December 1997. 2 E-mail: csyu@math.cycu.edu.tw.

0096-3003/98/$19.00 © 1998 Elsevier Science Inc. All rights reserved. PII: S0096-3003(97) 10034-0

286 M.-P. Chen et al. / Appl. Math. Comput. 91 (1998) 285~96

o o

f (z) = z + Z a , z " , (1) n -2

which are analytic in the open unit disk

ou := {z: z c c and Izl < 1}.

Also let 5 p denote the class of all functions in ~4 which are univalent in ~//. We denote by Y*(a) and ~('(a) the subclasses of 5 P consisting of all functions which are, respectively, starlike of order a and convex of order a in ~//(0 ~< a < 1), that is,

5 p , ( a ) : = { f : S E S ~ a n d ~ R(f__~_)zs'(z) > a ( 0 ~ < a < l ; z E ~ / / ) } (2)

and

{ ( z f " ( z ) ' ~ > a (0~<a< 1;zESg)}. (3) aug(a):= f : f c S t a n d ~ R 1 + f ' (z) )

It follows readily from the definitions (2) and (3) that

f (z) C Yf(a) ¢=~ zf'(z) E 5t*(a) (0 ~< a < 1), (4)

whose special case, when a = 0, is the familiar Alexander theorem (cf. , e.g. Ref . [5], p. 43, T h e o r e m 2.12) . W e note also that

Y(a ) CSP*(a) C 5 g (0~<a< 1), (5)

5~*(a) C_ 5~*(0) - ~* (0~<a < 1) (6)

and

x ( a ) c ~c(0) = x (0 ~< a < 1), (7)

where 5 #* denotes the class of all functions in d which are starlike (with res- pect to the origin) in ~//.

In statements like those involved in the definitions (2) and (3), and in anal- ogous situations throughout this paper, it should be understood that functions such as (see also Ref. [19])

zf'(z) and zf"(z) f (z) if(z) '

which have removable singularities at z = 0, have had these singularities re- moved.

For functions f (z) of the form (1) and belonging to the classes 5P*(a) and #C(a) defined by Eqs. (2) and (3), respectively, it is well known that (cf. Ref. [14])

M.-P. Chen et al. I Appl. Math. Comput. 91 (1998) 285~96

f E ,~*(~) ~ la.I <~ H~=20 - 2a) ( n - 1)!

and

2~)

n{ where, as usual,

2 8 7

( , ~ N\{1}) (s)

(n E N\{1}) , (9)

N := {1, 2, 3, . . .} and N 0 : = ~ U { 0 } .

In view of the second inclusion relation in Eq. (5), the special case of the asser- tion (8) when ~ = 0 is substantially weaker than de Branges' result (cf. Ref. [4]):

f e ,~'J(~ J*) ~ la, I <<,n (n ¢ N\{1}), (10)

where the equality holds true for all integers n >/2 only if f (z) is any rotation of the (Koebe) function K(z) defined by

2 z nz" (z E ~1[). (11) K(z) .-- (1 - z) 2 -- ,,_,

O n the other hand, in its special case when ~ = 0, the assertion (9) immedi- ately yields LSwner's result (cf. Ref. [7])

. f E , g = c l a , [ < ~ l (hE ~ \{1}) , (12)

where the equality holds true for all integers n/> 2 only if f (z) is any rotation of the function L(z) defined by

~ z " (z E 4/). (13) Z

L(z) := l - z = n 1

By applying the assertions (10) and (12), it is fairly straightforward to derive the following familiar growth-and-distortion type results for the function class- es ,~ and ,~', respectively (cf., e.g. Refs. [5,6]).

Theorem 1. I f the Jimction f(z) is in the class ~9 "~, then

[f(.)(z)l< ~ n!(n+ Izb) (ze~Y;n E No), (14) (1 - I~1 ) "+2

where the equality hoMs true for the Koebe function K(z) given by Eq. (11).

T h e o r e m 2. I f the function f (z) is in the class ,~, then

n! If(")(z)l ~< (~ e ,~; ,, ~ ~o),

(1 - Izl)

where the equality holds true for the function L(z) given by Eq. (13).

(15)

288 M.-P. Chen et al. I Appl. Math. Comput. 91 (1998) 285~96

Recently, Owa et al. [10,13] conjectured the existence of growth and distor- tion results of the types (14) and (15) involving the fractional derivative oper- ator Dj+~(n E N0; 0 ~< 2 < 1) given by Definition 2. However, these conjectures were shown to be false by Cho et al. [2] and Chen et al. [1], who did indeed prove generalizations of Theorems 1 and 2, respectively, in terms of fractional derivatives. Subsequently, Owa [9], Choi [3], and Srivastava [17] presented a number of growth-and-distortion type results relevant to (or extending) the works of Cho et al. [2] and Chen et al. [1]. The main objective of this paper is to present a further generalization of these growth-and-distortion type re- sults involving a novel class of analytic functions in 4l. We also show how the main results obtained in this paper are related to those given by the afore- mentioned researchers on the subject.

2. Operators of fractional calculus

In our present investigation, we shall make use of certain operators of frac- tional calculus (that is, fractional integral and fractional derivative). From among the numerous operators of fractional calculus (which have indeed been studied in the mathematical literature in one context or the other), we choose to recall here the fractional calculus operators given by Definitions 1 and 2 (cf. Ref. [8]; see also Ref. [12]):

Definition 1 (Fractional integral operator). The fractional integral of order 2 is defined, for a function f(z), by

1 f : f ( ( ) d( (2 > 0), (16) L)~;f(z) := F(2) (z Z - ~ 1-~

where f(z) is an analytic function in a simply-connected region of the z-plane containing the origin, and the multiplicity of (z - ~)x-I is removed by requiring log (z - ~) to be real when z - ~ > 0.

Definition 2 (Fractional derivative operator). The Jhactional derivative of order 2 is defined, for a function f(z) , by

_d_[z O~f(z) := F ( 1 - 2 ) d z J 0 ~ (17)

ff~'r D~ nf(z)

where f(z) is constrained, and the multiplicity of (z - ~)-~ is removed, as in Definition 1.

( n < ~ < n + 1;n¢ ~),

M.-P. Chen et al. / Appl. Math. Comput. 91 (1998) 285-296 289

We shall also require, in our study of the function class stemming essentially from the classes 5 P and ~f defined above, the Gaussian case

f - l = m = l

of the generalized hypergeometricfunction eFm given by the following definition.

Definition 3. Let )oi(J = 1, . . . ,g) and # j ( j = 1 . . . . . m) be complex numbers such that

/~i # 0 , - 1 , - 2 , . . . , 0"= l , . . . , m ) .

Then the generalized hypergeometric function ~.Fm(z) is defined by

fE,,(z) - fFm()~,,... ,2,.;pl,... ,/~m; Z)

= fErn z

L/~1 . . . . , ~,,;

.r~ ()q),,...()4), z~ ( e~<m+l ) , (18) : : ( ' , ) . (urn). n!

where (2), tions by

F(2 + n) { 1

( 2 ) , . - C(2) - - _ 2 ( 2 + 1 ) . . . ( 2 + n - 1 )

We note in passing that

z~'Fm()LI, " '" , ) ~ ; / / 1 , • • " , ]~m; z) E ,~:J,

denotes the Pochhammer symbol defined, in terms of Gamma func-

(n = 0;2 7~ 0), (19)

(20)

since the rFm series in Eq. (18) converges absolutely for (cf., e.g. Ref. [16], Ch. 2)

(i) [z] < oc i f g < m + 1; (i i) z E °h' i f g = m + 1; (iii) z E O°k" :=- {z: z E C and [z I = 1 } if g = m + 1, provided further that

~ / - 2j > 0, (21) j = l ]

unless (of course) the series terminates. In terms of the Gaussian case

g - l = m = l

of the generalized hypergeometric function (18), we now recall the definitions of some interesting generalizations of the fractional calculus operators given by Definitions 1 and 2.

290 M.-P. Chen et aLI Appl. Math. Comput. 91 (1998) 285-296

Definition 4 (GeneralizedJkactional integral operator). Under the hypotheses of Definition 1, the generalized .fractional integral of order 2 is defined, for a function f(z), by

._z,--,,c ( !) So, :x',''f(z) . - F(2) Jo (z-~)~-'2F, 2 + p , - v ; ) 4 1 - f (~)d~

(2 > 0; ~c > max{0,/x - v} - 1 ),

provided further that

f(z) = O(Izl ~) (z --+ 0).

It follows readily from Definitions 1 and 4 that

D?;LY(z) x'-;'" = s , ,~ .f(z) (;~ > o).

Furthermore, since

2Fl(a,b;b;z)=iFo(a; ; z ) = (1 - z)-" (z E ql),

we have the relationship:

2.p,-2 Io,: f(z) =DT:;z ; ;'f(z) (2 > 0).

(22)

(23)

(24)

(25)

(26)

l;t,p.v The operator *0,_~ is a generalization of the fractional integral operator which was studied by Saigo [15] and applied subsequently by Srivastava and Saigo [20] in solving various boundary value problems involving the Eule> Darboux equation

o,) OxOy x - y \ Ox = 0 ( ~ > 0 ; f l > 0 ; : c + f l < 1). (27)

Definition 5 (Generalized fractional derivative operator). Under the hypotheses of Definition 2, the generalized fractional derivative of order 2 is defined, for a function f(z), by

1 d ~ , g - - . _~ ( ,

F(1-- ) . )dz z - Jo ( z - g ) 2F. t P - z , - v ; 1 - ~ ;

Jo.z f(z) = f(~)d~ (0~<2 < 1), (2S)

d" x , , , -dTzJo;..7(z) ( n . < ; ~ < , + l ; n ~ ) (~c > m a x { 0 , # - v - 1} - 1),

where ~c is given, as before, by the order estimate (23).

M.-P. Chen et al. / Appl. Math. Comput. 91 (1998) 285-296 291

It follows readily from Definition 5 that

Jd".j"Vf(z) : D~f(z) (0 ~< 2 < 1), (29)

where the fractional calculus operator D~ is, in fact, given by Definitions 1 and 2 for all values of 2 (see e.g. Ref. [18], p. 343). Furthermore, in terms of Gamma functions, we have (cf. Refs. [11,21])

jo.,U,,,zp F ( p + I ) F ( p - u + v + 2 ) z"-" = F ( p ~ - ~ l ~ ( p - 2-~- v ~- 2)

(0~<2< 1 ; p > m a x { O , / ~ - v - 1 } - l ) . (30)

3. Bounds for a general class of analytic functions

By appealing appropriately to the familiar Euler transformation [16], p. 10, Eq. (1.3.15):

2F1 (a, b; c; z) =(1 - z)"-a-b2F, (c - a, c - b; c; z)

([ a r g ( l - z ) l ~< re - e(0 < e, < ~); c ¢ 0 , - 1 , - 2 , . . . ) (31)

as well as the Chu-Vandermonde summation theorem [16, p. 243, Eq. (III.4)]:

2Fl ( _ n , b; c; l ) _ ( c - b ) . (n E N0;c ¢ 0 , - 1 , - 2 , . . . ) , (32) (c),,

where (2), is defined, as before, by Eq. (19), it is not difficult to verify that

2 F l ( - n , ~ ; q ; o g ) > 0 (n E N0; t / > max{0,~};0~<o~< 1). (33)

This last positivity result (33) leads us naturally to a novel subclass (given by Definition 6) of the general class ,4(p) consisting of functions f ( z ) of the form:

,t>c

f ( z ) = ~ a , z " (p E N; ap 7 ~ 0), (34) n=p

which are analytic in the open unit disk 41. Clearly, we have [cf. Eq. (1)]

= (35)

where it is understood that a~ = 1.

Definition 6. A function f ( z ) given by Eq. (34) and belonging to the class ,~'(p) is said to be in the class ~ffp(~O; ~, q) if there exist real numbers 4, ~/, and ~ such that

. r E ~ p ( 6 ' ) ; ~, ?/) z:~ Jan[ ~ 2F 1 ( l - ?2, ~; ?/; fJ))

( n = p , p + l , p + 2 , . . . ; p E N;q>max{0 ,~} ;0~<m~<l ) , (36)

2 9 2 M.-P. ('hen et aL I AppL Math. Comput. 91 (1998) 285-296

where 2F~ is the Gauss hypergeometric function corresponding to the case

g - l = m = l

of Definition 3.

In view of the Chu-Vandermonde summation theorem (32), it is easily ver- ified that

4¢p(1;r / - ~,r/) = C¢,p({,q) CO E N;¢,q E [~+), (37)

where ~p(~, r/) denotes the subclass of ,e/Co ) considered recently by Srivastava [17]. In particular, it is not difficult to observe from Definition 6, and from Eqs. (8)-(10), that the class Yp({, r/) (that is, the class ..,'tip(l; q - {, q)) is anal- ogous to

(i) the class ST* (~) when

p = 1, { = 2 - 2 c ¢ and q = 1

(ii) the class .X/(~) when

p = l , ~ = 2 - 2 ~ , and r / = 2

and (iii) the familiar class 5 ~ itself when

(0~<c~ < 1);

(0~< :~ < 1)

p = 1, ~ = 2 , and q = 1.

By appealing to definition (34) and the assertion (36), we now prove our main result contained in what follows.

Theorem 3. I f the Junction f ( z ) is & the class Jfp(~O; ~, r/), then

n+;..t~.v . /a)(&*la,)) (,~ v; IJo= j (z) I ~< v , ,..,Ft, Izl)

(k + n + p) !F(k + n + p - It + v + 2) + ~ r(~ ~p-~ 7 ~ ~-U- ~ ~ ~ 72)

x 2F~ (1 - k - n - p , ~; q ; o~)Izl k+p- '*

(0 < ]z] < 1;n E N0;p E [N;0~<2 < min{1 ,v+ 2};

/t < min{1,v + 2};q > max{0, ~}; 0~<a)~< 1),

where

(38)

M.-P. Chen et al. I Appl. Math. Cornput. 91 (1998) 285 296

n-1

k=0

(k + p)!r( l~ + p - ~ + v + 2) x

/ r ( k + p - n - u + 1 ) l r ( k + p - ,~ + v + 2)

the sum being assumed to be nil for n -- O.

293

izi k+p n -g

(39)

Proof. First of all, by appealing to Definition 5 as well as the generalized fractional derivative formula (30), we find from Eq. (34) that

~X3

rn+},,lt,l'x-i , = d ' , ~ , u . V Z a f J O.z , l ~ Z ) d z n ~ O,z

k p

k ! C ( k - # + v + 2) ~_ . ,,

(0< Izl < 1 ;nCN0;p6 N;0~,<)~<min{1,v+2};

# < min{1, v + 2}). (40)

Since f E ,~p(e); 4, ~/), we can apply the assertion (36). Eq. (40) thus yields

iJo3>-,,,,,f(z)l ~ k ! r ( k - X + ~ + 2) izl~_,, ,,

~< ~-~ 2F~(1- k, ~ , ~ ; ~ ) i F ( k _ n _ v + 1 ) l / . ( k _ ? + v + 2 ) k=p

£ =O~'";°~(2, p,v;Izl)+ 2F~(1-k-p,~;~l;co) k -n

(k + p)!r(k + p - # ~ + v + 2) × Iz ( +~ . - , , I r ( k + p - n - ~ + 1 ) l r ( k + p - ;. + v + 2)

(0 < Izl < l;n E N0;p E N;0~<2 < min{1,v + 2};

/~ < min{1,v + 2}; q > max{0, ~}; 0~<co~< 1) (41)

where O~'";~)/(2, #, v; Izl) is given by Eq. (39). Finally, by an obvious shift of the index of summation, the last sum in

Eq. (41) is rewritten precisely as the sum occurring in the assertion. This evi- dently completes the proof of Theorem 3. []

By applying the relationship (29) between the fractional derivative operators Jo/z'' and D~, the special case of Theorem 3 when/~ = 2 immediately yields the following theorem.

294 M.-P. Chen et al. / Appl. Math. Comput. 91 (1998) 285-296

Theorem 4. I f the Jhnction f ( z ) is in the class dq~p(a~; ~, r/), then

D~+~[ z

(k + n + p ) ] + ~ c ( ; ~--p--- ~-~ l) zF,(1 - k - n - p , ~; . ; ~o)lzl k÷"-"

(0 < Jz] < 1;n E t~0;p E ~;0~<2 < 1;

q > max{0, ~};0 ~< ~ ~< 1)

where

"-' (k + p)! cbl,¢'";'") (2; [z[) :=~-~ zFl (1 - k - p, ~; q; w) [F(k + p - n - ), + 1)[

k=0

(42)

jzjk+p-°-q

(43)

the sum being assumed to be nil f o r n = O.

We find it to be worthwhile to record here yet another growth-and-distor- tion type result for the class ,~¢p(a~; ~, t/), which unifies and extends a number of known results (including, for example, the assertions of Theorems 1 and 2). Indeed, by setting 2 = 0 in Theorem 4 (or, alternatively, by setting /~ = 2 = 0 in Theorem 3), we obtain the following theorem.

Theorem 5. I f the function f ( z ) is in the class ,~p(a~; ~, q), then

If(') (z)[ ~< 7J~;'"~"/(]zl) + n! 2El (1 - k - n - p, ~; q; co) z k+p k=0

(z c ~//;n E t%;p 6 ~ ; t / > max{0, ~};0 ~< a~< 1), (44)

where

7J~¢'":"~)([zl) := n! 2F,(1 -k-n-p,¢;q;~o)lzl *+~-", (45) k=0

the sum being assumed to be nil f o r n = O.

Various known upper bounds for the fractional derivatives of functions be- longing to such simpler classes of analytic functions as

given earlier by (for example) Srivastava [17], Owa [9], Choi [3], Cho et al. [2], and Chen et al. [1], would follow from one or the other of Theorems 3, 4, and 5 above by suitably specializing the parameters involved. We choose to leave the details of these derivations as an exercise for the interested reader.

M.-P. Chen et al. / Appl. Math. Comput. 91 (1998) 285~96

Acknowledgements

295

T h e p re sen t i nves t i ga t i on was c o m p l e t e d d u r i n g the s e c o n d a u t h o r ' s visi ts to

the Ins t i t u t e o f M a t h e m a t i c s ( A c a d e m i a Sinica) at Ta ipe i , T a m k a n g U n i v e r s i t y

at T a m s u i , S o o c h o w U n i v e r s i t y a t Ta ipe i , N a t i o n a l Ts ing H u a U n i v e r s i t y at

H s i n - C h u , a n d C h u n g Y u a n C h r i s t i a n U n i v e r s i t y at C h u n g - L i in D e c e m b e r

1996 a n d J a n u a r y 1997. Th i s w o r k was s u p p o r t e d , in par t , by the N a t i o n a l Sci-

ence C o u n c i l o f the R e p u b l i c o f C h i n a u n d e r G r a n t N S C - 8 6 - 2 1 1 5 - M - 0 0 1 - 0 0 1

and , in par t , by the N a t u r a l Sciences a n d E n g i n e e r i n g R e s e a r c h C o u n c i l o f

C a n a d a u n d e r G r a n t O G P 0 0 0 7 3 5 3 .

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