A STUDY ON A CLASS OF $p$-VALENT FUNCTIONS ASSOCIATED WITH GENERALIZED HYPERGEOMETRIC FUNCTIONS Текст научной статьи по специальности «Математика»

Владикавказский математический журнал 2015, Том 17, Выпуск 1, С. 31-38

A STUDY ON A CLASS OF p-VALENT FUNCTIONS ASSOCIATED WITH GENERALIZED HYPERGEOMETRIC FUNCTIONS1

E. El-Yagubi, M. Darus

In this paper, we study and introduce the majorization properties of a new class of analytic p-valent functions of complex order defined by the generalized hypergeometric function. Some known consequences of our main result will be given. Moreover, we investigate the coefficient estimates for this class.

Mathematics Subject Classification (2000): 30C45.

Key words: majorization, p-valent functions, hypergeometric functions.

1. Introduction

Let Ap be the class of fun étions f (z) normalized by

f (z) = zp + ^ ap+„zp+n, p G N, (1.1)

n= 1

whieh are analytic and p-valent in the unit disc U. Let f and g be analytic in the open unit disc U. We say that f is majorized by g in U and write

f (z) « g(z) (z G U), (1.2)

if there exists a function analytic in U such that

№(z)| < 1, f (z) = p(z)g(z) (z G U). (1.3)

It may be noted here that (1.2) is closely related to the concept of quasi-subordination between analytic functions.

f(z) g(z) U, f g

Schwarz function o>, analytic in U, with o>(0) = 0 and |o>(z)| < 1 such that f (z) = g(o>(z)), z G U. We denote this subordination by f (z) X g(z). If g(z) is univalent in U, then the subordination is equivalent to f (0) = g(0) and f (U) C g(U).

If f (z) and g(z) belong to Ap, then the Hadamard product f * g is defined by

f (z) * g(z) = zp + ap+nbp+nzp+n, p G N.

n=1 p

hypergeometric functions

r%(al,bl;zn = zp + jr \n'• ' ' ^, P G N,

^-f (bi)n ••• (bs)n n!

n=1

© 2015 El-Yagubi E., Darus M.

1 The work presented here was partially supported by UKM grant AP-2013-009.

where a G C, bq G C\{0, -1, -2,...}, (i = 1,..., r, q = 1,..., s), and r ^ s + 1; r, s G No, and (x)n is the Poehhammer symbol defined by

_ T{x + n) _ fl, n = 0,

r(ir) ~ \x(x + l)---(x + n-l), n = { 1,2,3,...}.

Let 2 p G Ap is defined by

L mb = zp + Y^

LAl ,A2 ,p = z +

n=1

p + (Ai + A2)n + b

p + A2n + b

m

p G N,

where m, b G No = N U {0} A2 ^ Ai ^ 0.

Corresponding to r(Ss(a1 ,b1;zp), 2 p and using the Hadamard product, we define a

new generalized differential operator D^m bA2 p(a1,b1) as follows:

Definition 1.1. Let / £ Ap, then a generalized differential operator D^m ba2 p(a1,b1)/(z) :

Ap ^ Ap is given as

Dml'p(«1,b1)/(z) = (rGs(a1 ,b1; zp) * L^* /(z))

= zp

+

p + (A1 + A2)n + b

n=1

It follows from the above definition that

p + A2n + b

(ai)ra ■ ■ ■ («r)ra Qp+raZP+" (L4)

(6l)„... (bs)n n\

(p + A2n + b) D^bp(a1,b1)/(z)

= (p + A2n - pA1 + b) Dm;bA2'p(a1 ,b1)f (z) + A^Dm^>1,b1)/(z))'.

(1.5)

Remark 1.1. It should be remarked that the linear operator Dm' (a1,b1)/(z) is a

generalization of many operators considered earlier. Let us see some of the examples:

A2 = b = 0, and Karthikeyan [1].

For A2 = b = 0, the operator Dm ' b2 p(o1, b1)/ reduces to the operator was given by Selvaraj

For m = 0, the operator Dm' p(a1,b1)/ reduces to the operator was given by El-Ashwah [2].

For m = 0 and p = 1, the operator Dm' p(o1 ,b1)/ reduces to the well-known operator introduced by Dziok and Srivastava [3].

A2 = b = 0 p = 1,

For m = 0 r = 2, s = 1 and p = 1, we obtain the operator which was given by Hohlov [5].

For r = 1, s = 0 a1 = 1 A1 = 1, A2 = b = 0 and p = 1, we get the Salagean derivative operator [6].

For r = 1, s = 0 a1 = 1 A2 = b = 0 and p = 1, we get the generalized Salagean derivative operator introduced by Al-Oboudi [7].

For m = 0 r = 1, s = 0 a1 = S + 1 and p = 1, we obtain the operator introduced by Ruseheweyh [8].

For r = 1, s = 0 a1 = S + 1 and p = 1, we obtain the operator studied by El-Yagubi and Darus [9].

For m = 0 r = 2 and s = 1, a2 = 1 and p = 1, we obtain the operator studied by Carlson and Shaffer [10].

For r = 1, s = 0 a1 = 1, A2 = 0 and p = 1, we get the operator introduced by Catas [11]. Next, by using the generalized differential operator D""' \2 p(a1,61), we study the class aj'p[ai, bi,A, B,7] as follows:

Definition 1.2. Let f g ap, then f g S"a1'a2 p[a1,b1, A, £,7] of p-valent functions of complex order 7 = 0 in U, if and only if

' 1 (z^D^oMKz^ \\ 1 + Az 1 H— ---p + 1 \ > -<-, z £ U,

(1.6)

where p G N m, b, j G N0 = N U {0} 7 G C\{0}, A2 ^ A1 ^ 0 -1 < B < A < 1, a» G C, bq G c\{0, -1, -2,...} (i = 1,... ,r, q = 1,... ,s) and r ^ s + 1 r, s G N0.

Clearly, we have the following relationships:

(i) when m = 0 p = 1, j = 0 r = 2, s = 1 a1 = b15 a2 = 1 A = 1 and B = -1, then the class S""'Aj p[a1, b1, A, B, 7] reduces to the class S(7).

(ii) when m = 0 p = 1 j = 1 r = 2 s = 1, a1 = b15 a2 = 1, A = 1 and B = -1, then the class S""' Aj p[a1, b1, A, B, 7] reduces to the class C(7).

(iii) when m = 0 p =1 j = 0 r = 2, s = 1 a1 = b15 a2 = 1 A = 1 B = -1 and 7 = 1 - a, then the class S"1' Aj p[a1, b1, A, B, 7] reduces to the class S*(a) for 0 < a < 1.

The classes S(y) and C(7) are said to be classes of starlike and convex of complex order 7 = 0 in U, were considered by Nasr and Aouf [12] and S*(a) denote the class of starlike aU

2. Majorization Problem

a majorization problem for functions f belong to the class S"1' Aj p[a1,b1, A, b,y] is

considered. Theorem.

Let f G Ap and suppose that g G S"1' Aj p[a1,b1, A, b,y]. If

(D"' a2,p(a1,b1)f (z)) JS majorized by (D"1'A2,p(a1,b1)g(z))(j) i« U, then

Ai' A2 p

(DSA+1;bp(a1,b1 )f (z))(j) < (dA+a1;bp(a1,b1 )g(z))(j for |z| < ro

Ai; A2 ,p(a1,

(2.1)

where r0 = r0(p, 7, A1, A2, b, A, B) is the smallest positive root of the equation

7(A-£)+(j? + A2W + 6)£ -

+ 2

A1

p + A2n + b

A1

+ 2|B|

r+

p +A2n + 6 Ai

(2.2)

-1 < B < A < 1; A2 ^ A1 ^ 0; b G N0; P G N; 7 G C\{0}.

< Sinee g G S" ' Aj p[a1 ,b1, A, b,y] we can get from (1.6), that

1 1 (W^MM*))^ \_1 + Aw(z)

A {D™;bMiP{aiM)g{z))^ P J) 1 + Bw(z)'

(2.3)

3

2

r

r

0

(2.4)

where 7 G C\{0}, j, p G N p > j and w is analytic in U with

w(0)=0, |w(z)| < 1 (z G U).

From (2.3), we get

z(Dx;hM,P(.aMg{z)){3+1) = (p-j) + [^A-B) + ip-j)B]w(z)

By noting that

and by virtue of (2.4) and (2.5) we get

(Dml ' p(01 ,b1)g(z))(j)

p+\2n+b r-, , |n| ||

^ --*-Al [ ' "j-*-1 I • (2-6)

(2.5)

( p+X2n+b \

V Al )

-.m,' b / 7 \ j-/ \\(j) ■ • ■ 1 i i r^m 'b

Next, since (Dm' a2 p(a1 ,b1)f (z))(j) is majorized by (Dm' a2 p(a1,b1 )g(z))(j) in the unit disc U, thus from (1.3) we have

(Dm'bA2>1 ,b1)/(z))(j) = ^(z)(Dm'bA2'p(a1 ,b1)g(z))(j).

zz

z(Dm'bA2' p(01,b1)/(z()(j+1) = zp' (z)(Dm;bA2 'p(a1,b1 )g(z))(j) + z^(z)(Dm'bA2' p(01, b1)g(z))(j+1). Now by using (2.5) in the above equation, it yields

№1' bp(01,b1)/(z))(j) zp'(z)(DmbA „(01 ,b1)g(z))(j) + b ... (2.7)

Ai

Thus, by noting that p G ^ satisfies the inequality

{z£ U), (2.8)

by using (2.6) and (2.8) in (2.7), we get

(DmS bp(01,b1)/(z))(j) 1 - |p(z)|2 |z|(1 + |B||z|)

|p(z)| +

l-\z\2 + \B\\Z\

l^'bp(01 ,b1)g(z))(j) |, (2.9)

which upon setting

leads us to the inequality

<

|z| = r and |^>(z)| = p (0 ^ p ^ 1)

(da+a!' bp(a1,b1 )f (z))(j) 0(p)

(1 - r2)

( p+X2n+b \

V Al )

7 (A-B)+ (P+x^+AB

(d"X bpg(z))(j)

(2.10)

where

0(p) = r(1 + |B|r)p2 + (1 - r2)

p + X2n + b

Ai

Y(A - B) +

p + X2n + b

Ai

B

p + r(1 + |B |r)

(2.11)

takes its maximum value at p = 1 with r0 = r0(p, y, A1 ,A2,b, A, B). Here r1(p,Y, A1, A2,b, A, B) is the smallest positive root of the equation (2.2).

Furthermore, if 0 ^ p ^ r1(p, y, A1,A2, b, A, B), then the function ^(p) defined by

^p) = -a(l + \B\a)p2 + (l-a2)\

A1

A1

p + a(1 + |B|a)

(2.12)

is seen to be an increasing function on the interval 0 ^ p ^ 1, so that

V(p) < ^(1) = (1 - a2)

P + ^n + b\_ l{A_B)+(P±>^±b]B(T

A1

A1

(2.13)

0 < p < 1 (0 ^ a ^ ri(p, y, Ai,A2,b,A,B)).

p=1

|z| ^ r1(p, Y, A1, A2,A, B) where r1(p, y, A1, A2,A, B) is the smallest positive root of equation (2.2).

Putting A = 1 and B = -1 in Theorem 2.1, we have the following result: Corollary 2.1. Let f G Ap and suppose that g G S""'Ajp(a1 ,b1,Y). If (D"1 p(a1,b1 )f (z))(j) is majorized by (D™ p(a1, b1)g(z))(j) in U, then

(DA+p1(ai,bi)f(z))(j) < (DA+1 (ai,bi)g(z))^ for |z| < r0

(j)

(2.14)

where

r0 = r0(p,Y, Ai, A2, b) =

k-^k2- 4(P+x^+b) |27 - (P+x^+b)

2|27- (p+AA2"+b)

fe = 2+(^+AA2W + 6) +

2Y -

p+ A2n + b Ai

p G N; Y, Ai G C\{0}, b G N0, A2 ^ 0

and S"'Aj p(a1, b1, y) be a special case of S™' A2 p[a1,b1, A, B, y] when A = 1 and

A b ; j

B = -1.

r

x

r

X

Setting p = 1, m = A2 = b = 0 A1 = 1, j = 0 r = 2 s = 1 a1 = b1 and a2 = 1 in Corollary 2.1, we get the following corollary:

Corollary 2.2. Let / G Ap and suppose that g G S(y). If /(z) is majorised by g(z) in U, then

|/'(z)| < |g'(z)| (|z| <r3),

where

3+|27-l|-V9 + 2|27-l| + |27-lF 0 °U) 2|27-1|

which is a known result obtained by Altintas et al. [13].

For 7 = 1, the Corollary 2.2 reduces to the following result:

Corollary 2.3. Let /(z) G Ap and suppose that g G S* = S*(0). If /(z) is majorized by g(z) in U, then

\f'(z)\^\gf(z)\ (|z|< 2-V3), which is a known result obtained by MacGregor [14].

3. Coefficient Estimates

The coefficient estimate for the class Sm'Aj p[a1, b1, A, B, 7] is obtained, when j = 0.

Definition 3.1. Let Sm'A2 p[a1, b1, A, B, 7] denote the subelass of p-valent functions which satisfy the condition

! | 1 (z(D^2,P(aMf(z)y \ 1 + Az l{ D^p{aiM)f{z) V 1 + Bz'

where p G N Y G C\{0} A2 ^ A1 ^ 0 m, b G No, -1 < B < A < 1 a G C, bq G C\{0, -1, -2,...}, (i = 1,... ,r,q = 1,...,s)^dr ^ s + 1 r, s G No.

Theorem 3.1. Let / G Ap. If it satisfies the condition:

IY I(A - B )

< 1, (3.2)

then / G SAi ' a2 p

< Let / G Sm' A2 p[a1,b1, A, B,y], then we can write (3.1) as follows:

x 1 \_i + Aw(z)

n D^X2,p(a1,b1)f(z) V) 1 + Bw(z)

which gives

x2,P(ai,bi)f(z)y D7;bx2,P(*Mf(z)

p

/z(Dm 'bA n(01,b1)/(z))' 1(A-B)-B\ Xl'X2'p -v

V D^iaMHz) ,

w(z). (3.3)

From (3.3), we obtain

pzp + E

n=1

p+(Ai+A2)ra+b

p+\2n+b

+ E

n=1

p+(Ai+A2)ra+b

P+A2ra+6

p

Y (A - B) - B

pzp + E

n=1

p+(Ai+A2)ra+b p+A2ra+6

<x

+ E

n=1

p+(Ai+A2)ra+b P+A2ra+6

(11 )n'"K)ii ■>•£>+«,

-p

w(z)

which yields

E n

n=1

p+(Ai+A2)ra+b p+A2ra+6

(«lV-(%)n fi

<x

1+ E

n=1

p+(Ai+A2)ra+b p+\2n+b

(ll)n'"K)n ,, fi

y(a - B) - B

n E

n=1

p+(Ai+A2)ra+b P+A2ra+6

(ll)n'"K)n -n

1+

n=1

p+(Ai+A2)ra+b p+A2ra+6

(ll)ll'"K)n r;

(bi)„-(ba)„n\aP+™Z

w(z).

Since \w(z)\ ^ 1,

£

n=1

n

p + (A1 + A2 )n + b

p + A2 n + b

(ai)ra • • • (flr)ra „

(b1)n ••• (bs)nn!

y(A - B) [Bn - y(A - B)]

n=1

p+ (Al + X2)n + b p + A2n + b

(«l)ra ' ' ' («r)ra r;

Letting \z\ —y 1 through real values, we have

£[n + |y(A - B) - Bn|]

n=1

p+ (Al + X2)n + b p + A2n + b

(«l)ra ' ' ' («r)ra

(bi)n ■ ■ ■ (bs)nnl

|ap+n| < |y|(A - B),

therefore,

E^=1 [n + |Y(A - B) - Bn|]

p+(Ai+A2)ra+b p+\2n+b

(bi)n---(bs)nn\

\ 1

Iy\(A - B )

^ 1.

p

in [15, 16].

References

1. Selvaraj C., Karthikeyan K. R. Differential subordination and superordination for certain subclasses of analytic functions // Far East J. of Math. Sci.-2008.-Vol. 29, № 2.-P. 419-430.

2. El-Ashwah R. M. Majorization Properties for Subclass of Analytic p-Valent Functions Defined by the Generalized Hypergeometric Function // Tamsui Oxf. J. Math. Sci.—2012,—Vol. 28, № 4.—P. 395-405.

m

m

m

m

m

m

m

m

m

m

3. Dziok J., Srivastava H. M. Classes of analytic functions associated with the generalized hypergeometric function // Appl. Math. Comp.-1999.-Vol. 103, № l.-P. 1-13.

4. Selvaraj C., Karthikeyan K. R. Univalence of a general integral operator associated with the generalized hypergeometric function // Tamsui Oxf. J. Math. Sci.—2010—Vol. 26, № 1,—P. 41-51.

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8. Ruscheweyh S. New criteria for univalent functions // Proc. Amer. Math. Soc.—1975.—Vol. 49.— P. 109-115.

9. El-Yagubi E., Darus M. A new subclass of analytic functions with respect to fc-symmetric points // Far East J. of Math. Sci.-2013.-Vol. 82, № l.-P."45-63.

10. Carlson B. C., Shaffer D. B. Starlike and prestarlike hypergeometric functions // SIAM J. Math. Anal.— 1984.—Vol. 15, № 4.-P. 737-745.

11. CMAs A. On certain class of p-valent functions defined by a new multiplier transformations // Proceedings Book of the International Symposium G. F. T. A.—Istanbul: Istanbul Kultur University, 2007.—P. 241-250.

12. Nasr M. A., Aouf M. K. Starlike function of complex order // J. Natur. Sci. Math.—1985.—Vol. 25, № l.-P. 1-12.

13. Altintas O., Ozkan O. and Sri v.astava H. M. Majorization by starlike functions of complex order // Complex Variables Theory Appl.-2001.-Vol. 46, № 3.-P. 207-218.

14. MacGregor T. H. Majorization by univalent functions // Duke Math. J.—1967.—Vol. 34.—P. 95-102.

15. Darus M., Ibrahim R. W. Multivalent functions based on a linear operator // Miskolc Math. Notes.— 2010.-Vol. 11, № l.-P. 43-52.

16. Ibrahim R. W. Existence and uniqueness of holomorphic solutions for fractional Cauchy problem // J. Math. Anal. Appl.-2011.-Vol. 380.-P. 232-240.

Received September 16, 2014-

Entisar El-Yagubi

School of Mathematical Sciences

Faculty of Science and Technology

Universiti Kebang Set till Malaysia

Bangi 43600 Selangor D. Ehsan, Malaysia

Email:entisar_el980<3yahoo. com

Maslina Darus

School of Mathematical Sciences

Faculty of Science and Technology

Universiti Kebang Set till Malaysia

Bangi 43600 Selangor D. Ehsan, Malaysia

Email:mas 1 inaOukm. edu. my

ИССЛЕДОВАНИЕ ОДНОГО КЛАССА р-ВАЛЕНТНЫХ ФУНКЦИЙ, ПОРОЖДЕННОГО ОБОБЩЕННОЙ ГИПЕРГЕОМЕТРИЧЕСКОЙ ФУНКЦИЕЙ

Эль-Ягуби '-).. ,Чарус М.

Изучаются свойства мажорации для нового класса аналитических р-валентных функций комплексного порядка, порожденного гипергеометрической функцией. Приводятся некоторые известные следствия полученных результатов. Дяны также оценки коэффициентов для этого класса.

Ключевые слова: мажорация, p-валентная функция, гипергеометрическая функция.