On certain classes of fractional p-valent analytic functions Текст научной статьи по специальности «Математика»

Серия «Математика» 2015. Т. 11. С. 28—38

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ИЗВЕСТИЯ

Иркутского государственного университета

УДК 518.517

On certain classes of fractional p-valent analytic functions *

E. El-Yagubi

Universiti Kebangsaan Malaysia

M. Darus

Universiti Kebangsaan Malaysia

Abstract. The theory of analytic functions and more specific p-valent functions, is one of the most fascinating topics in one complex variable. There are many remarkable theorems dealing with extremal problems for a class of p-valent functions on the unit disk U. Recently, many researchers have shown great interests in the study of differential operator. The objective of this paper is to define a new generalized derivative operator of p-valent analytic functions of fractional power in the open unit disk U denoted by DT\1 a2 p af (z). This operator generalized some well-known operators studied earlier, we mention some of them in the present paper. Motivated by the generalized derivative operator DA^I\2 p af (z), we introduce and investigate two new subclasses SA^'\2 p a(f, v) and TSAA'' a2 p a(f,v), which are subclasses of starlike p-valent analytic functions of fractional power with positive coefficients and starlike p-valent analytic functions of fractional power with negative coefficients, respectively. In addition, a sufficient condition for functions f e £p ' a to be in the class SA' \2 p a(f,v) and a necessary and sufficient condition for functions f e Tp,a will be obtained. Some corollaries are also pointed out. Moreover, we determine the extreme points of functions belong to the class TSm'\2 p a(f, v).

Keywords: analytic functions, p-valent functions, starlike functions, derivative operator.

1. Introduction

Let denote the class of functions of the form

f (z) = zp+a + anzn+a, (z e U), (L1)

n=p+1

which are analytic in the open unit disk U = {z : z G C, \z\ < 1}, where p G N, a > 0, p > a, z G U.

* This work was supported by AP-2013-009.

For the Hadamard product or convolution of two power series f defined in 1.1 and a function g where

X

n+a

g(z) = zp+a + £ bnz

n=p+1

is

x

f (z) * g(z) = zp+a + £ anbnzn+a, (z e U).

n=p+1

We also denote by Tp,a the subclass of Ep,a consisting of functions of the form

X

f (z) = zp+a - \an\zn+a, (1.2)

n=p+1

which are analytic and univalent in the open unit disk U. For the Hadamard product or convolution of two power series f defined in (1.2) and a function g where

X

g(z) = zp+a - \bn\zn+a, (z e U)

n=p+1

is

f(z) * g(z) = zp+a - \an\\bn\zn+a, (z e U).

n=p+1

For a function f e Ep,a given by 1.1, we define the derivative operator

DXlM,p,a by

<12 ,pj(z) = zp+a + £

n=p+1

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

m

anzn+a, (z e U), (1.3)

where m,b e N0 = N U{0}, A2 > A1 > 0, p e N, a > 0,p> a.

Remark 1. It should be remarked that the differential operator vm;\2 p af (z) is a generalization of many operators considered earlier. Let us see some of the examples: for A1 = 1,A2 = b = 0,p = 1 and a = 0, we get the operator introduced by Salagean [11]. For A2 = b = 0,p = 1, and a = 0, we get the generalized Salagean derivative operator introduced by Al-Oboudi [1]. For A1 = 1,A2 = 0,p = 1 and a = 0, we obtain the operator introduced by Flett [7]. For A1 = 1,A2 = 0,b = 1,p = 1 and a = 0, we obtain the operator introduced by Uralegaddi and Somanatha [13]. For A2 = 0 and a = 0, we get the operator introduced by Catas [2]. For A2 = 0,A1 = 1 and a = 0, we get the operator introduced by Kumar et al. [9].

Clearly, by applying the operator Dm^\2 p a successively, we can obtain the following:

Dm,b f ( z) _ / ,A2 ,p,a D1 ,X2,p,a)f (z),m e N,

AlM'p'aJ ( ) = I f (z),m = 0.

f e Ep,a ^ D\i,\2,p,a e EP,a.

A function f e Ep,a is said to be in the class Pa(p,i), (0 < i < p + a) if and only if it satisfies the inequality

} >1 (z e U). M

The classes P0(1, 0) and P0(p, 0) were investigated in [10] and [12], respectively.

Now we define the subclass Smf\2 pa(l, v) of Ep,a consisting of functions of the form 1.1 and satisfying the analytic criterion

»{ZDMZf- - "} - p+a)i'u e u)'

,X2 ,p,aJ (Z) ,X2 ,p,aJ (Z)

(1.5)

where m,b e N0, Л2 > Ai > 0, p e N, a > 0,p > a, 0 < ц <p + a, v > 0.

2. Main results

We obtain a necessary and sufficient condition for functions f (z) G X

p,a ■

Theorem 1. Let f e Ep,a. A .sufficient condition for a function of the

form 1.1 to be in S^'\2 p a(ß,v) is that

E

n=p+1

p + (Ai + A2)(n - p) + b p + A2(n - p) + b

)(n - p)(1 + V) - (p + a - ] \an\ < 1, p + a - Ц.

(2.1)

where z e U, m,b e N0, Л2 > A1 > 0, p e N, a > 0,p > a, 0 < ц < p + a, v > 0.

ON CERTAIN ClASSES OF FRACTIONAL p-VALENT ANALYTIC FUNCTIONS 31 Proof. Let f be of the form 1.1. Our aim is to show that

Ax f(z))'

ft

Z(DA1',A2,p,a

D

•m'b f (z) A1'A2'PaJ (z)

f(z))' 1 z(Dm'b — ß]>v

D

— (p + a)

=> v

=> v

Z(.D\A2,p,a f (z))'

D

¡m' b

Ax' A2 'P' a m b

f (z)

Z(DmbA2, p,a f (z))'

D

m b

Ax'A2 P' a

< p + a — ß

m b

Ax'A2 p' a

v

DAm

f(z)

f(z))'

— (p + a) I —

— (p + a) I —

•m'b f (z) Ax' A2 p' af (z)

m'b f(z))'

zDm

' A2 p' a-l

D

m b

Ax'A2 p'a Z{D\!' A2 p' a

f (z)

f(z))'

D

m b

D

m b

Ax' A2 'P'a

< p + a — ß.

f(z)

— (p + a) I —

D

Ax'A2 p' a

f(z)

f(z))'

— ß} < 0 — ß^+p+a—ß

m b Ax A2 P a

D

m b

Ax A2 P a

f(z)

— (p +

Hence it suffices to prove that

ziDmbA2' P' af (z))'

D

m b A1 A2 P a

f(z)

— (p + a) I —

z{DmbA2' P ' af (z))'

f (z)

D

m b

— (p +

A1 A2 P a

< p + a — ß, (z £ U).

Yields

(2.2)

z(Dmx'' A2 p' a

f(z))'

D v

m b f(z)

Ax A2 P af(z)

z{DM' A2 P' a

— (p + a) I —

' z{Dm' A2 P'a

f(z))'

D

m,b

f(z))'

DX-!,A2'P'af (z)

= (1 + v)

— (p + a)

+

Ax'A2'P'af (z)

f(z))'

— (p +

z(Dm!,A2'P'a

<

z(DmbX2, Pa af (z))' — (p + a)Dm b

Dmx' A2 P' af (z) A1 A2 P a

f(z)

— (p + a)

D

m b

A1 A2 P a

f(z)

<

<

(1 + v) En=P+1(n — p) p+(Ax+A2)(n-p)+b P+A2 (n-p)+b m lanllzln+a

|z|P+a + E^P+1 p+(Ax+A2)(n-p)+b P+A2(n-P)+b m anllzln+a

<

<

(1 + v) En=p+l(n — p ) p+(Ax+A2)(n-p)+b p+A2(n-p)+b m lanl

1 + n=p+1 p+(Ax+A2)(n-p)+b p+A2(n-p)+b m lanl

v

where z ^ 1 along the real axis. This last expression is bounded by p+a -1 if

£

n=p+1

p + (Aj + Л2)(n - p) + b p + A2(n - p) + b

[(n - p)(1 + v) - (p + a - /)]\an\ < < p + a - (z £ U).

The proof of theorem 1 is complete. □

Next result describes the starlikeness for functions in the class (Spa(i,v)), which is an extension to the class of starlike functions defined by ¡3].

Corollary 1. Let the assumptions of theorem 1 hold. Then

»{ z(Dmb-paf (z>Y } >1.

D\1,\2,p,af (z)

Proof. By letting v = 0 in theorem 1, we obtain the desired result. □ We also introduce the class of starlike functions of fractional power of

order / (S*a(/)).

Corollary 2. Let the assumptions of theorem 1 hold. Then

*{ fZf}

Proof. By setting m = 0,v = 0 in theorem 1, we have the required result.

□

Finally, we have the next result which is an extension to the class of starlike function S*.

Corollary 3. Let the assumptions of theorem 1 hold. Then

*{ fzf } > 0

Proof. By setting m = 0,/ = v = 0 in theorem 1, we have the required result. □

Now we prove a sufficient condition for f £ Tp,a. Consider the subclass TSmьХъpa(/,v) of functions in Tp,a.

Известия Иркутского государственного университета. 2015. Т. 11. Серия «Математика». С. 28-38

m

Theorem 2. Let f be defined by 1.2. Then f £ TSmA p a(ß,v) if and only if the condition the following condition is satisfied.

E

n=p+1

p + (Xi + X2)(n — p)+ b p + X2(n — p) + b

\(p — n)v + (n + a — ß)] lanl < 1, p + a — ß

(2.3)

where z G U, m,b G N0, A2 > X1 > 0, p G N, a > 0,p > a, 0 < i < p + a, v > 0.

Proof. The sufficiency as in theorem 1, we need only to prove the necessity. Let f G TSm a2 p a(l,v) then we obtain

ft

zDm ' A2 p' a f(z))'

D v

m b f(z) Ax' A2 p' af (z)

zDmbA2, p' af (z))'

D

m b

Ax'A2 p'a m b

f(z)

— ß >v

— (p +

<Dm'bA2,p'af (z))'

D

m b

Ax'A2 p'a

f(z)

— (p + a)

ft

ziDm;bA2'p' af (z))' 'p'af (z))'

— ß — v [-

Dm.'f(z) Dm'b

Ax' A2 p' aJ

'\l, X2 p ,a

f(z)

> 0, (^(z) < \z\).

Thus when z ^ 1 along the real axis, we pose

— (p +

p + a — E n=p+1(n + a) p+(Ax+A2)(n-p)+b p+A2(n-p)+b m an

1 _ ¿—1 n=p+1 p+(Ax+A2)(n-p)+b p+A2(n-p)+b m an

ß

vE ~=p+1(p — n) p+(Ax+A2)(n-p)+b P+A2(n-p)+b m an

1 _ n=p+1 p+(Ax+A2)(n-p)+b P+A2(n-p)+b m an

0

p+a— ß—

n=p+1

p + (Ai + X2)(n — p)+ b p + X2(n — p) + b

(n + a — ß+v(p — n))an > 0,

and obtain the desired inequality: " p+(A1 + X2)(n—p)+ b

E

n=p+1

p + X2(n — p)+b Hence The proof of theorem 2 is complete

[(p—n)v+(n+a—ß)] an <p+a— ß, (z £ U).

□

m

Corollary 4. Let f be defined by 1.2 be in the class TS^^p a(p, v). Then we have

|an|<

p + a - л

p+(Al+A2)(n-p)+b p+\2(n-p)+b

[(p - n)v + (n + a - л)]

(2.4)

where m,b e N0, A2 > A1 > 0, p e N, a > 0,p > a, 0 < i < p + a, v > 0.

We shall now determine the extreme points of the class TSmi'\2 p a(l, v). Theorem 3. Let f (z) = zp+a and

fn(z) = ZP+a -

p + a - л

p+(Ai +\2){n-p)+b p+\2{n-p)+b

[(p - n)v + (n + a - /)]

where m,b e N0, A2 > A1 > 0, p e N, a > 0,p > a, 0 < i < p + a, v > 0. Then f e TSm,'bx2 p a(l, v) if and only if it can be expressed in the form

f (z) = Y1 Wnfn(z),

(2.5)

n= p

where Un > 0 and^X=p Un = 1. Proof. Suppose that

f (z) = wnfn(z) = Wpfp(z) + Y^ unfn(z) = Wpfp(z)+

n= p

n=p+1

+

n=p+1

Wn

zp+a-

p + a - л

p+(Ai +A2)(n-p)+b p+A2(n-p)+b

£

n= p

Wnzp+a- ^ Wn n=p+1

[(p - n)v + (n + a - /)]

p + a - л

p+(Ai+A2 )(n-p)+b p+A2(n-p)+b

[(p - n)v + (n + a - л)]

zp+a -

^ Wn n=p+1

p + a - л

p+(Ai+A2)(n-p)+b p+A2(n-p)+b

[(p - n)v + (n + a - л)]

m

m

m

m

Now

f (z) = zp+a -22 \an\ n=p+1

zp+a _

22 ^n

n=p+1

p + a — i

p+(\l+\2)(n-p)+b p+\2(n-p)+b

[(p — n)v + (n + a — ¡)]

therefore,

\an\ = Un

Now, we have that

p + a — i

p+(\l+\2)(n-p)+b p+\2(n-p)+b

[(p — n)v + (n + a — ¡¡)]

Setting

22 Un

n=p+1

22 Un = 1—Up < i.

n=p+1

p + a — i

p+(Xi +\2)(n-p)+b p+\2(n-p)+b

p+(Xi+X2)(n-p)+b p+\2(n-p)+b

[(p — n)v + (n + a — i)] [(p — n)v + (n + a — ¡¡)]

p + a — i

1,

we get

E

n=p+1

p+(Xi+X2)(n-p)+b p+X2(n-p)+b

[(p — n)v + (n + a — i)]

p + a — i

-\a,n\ < 1.

Therefore,

E

n=p+1

p + (A1 + \2)(n — p) + b p + X2 (n — p) + b

[(p — n)v + (n + a — i)] \an\ <p + a — i

It follows from theorem 2 that f G TSX^' X2 p a(i, v).

m

m

m

m

m

m

Conversely, we suppose that f e TSm±'\2 p a(l, v), it is easily seen that

f (z) = zp+a - £ \an\ n=p+1

zp+a -

Wn

n=p+1

p + a - л

p+(Ai+A2)(n-p)+b p+A2(n-p)+b

[(p - n)v + (n + a - л)]

which suffices to show that

|an| = Wn

p + a - л

p+(Ai+A2)(n-p)+b p+A2(n-p)+b

m b

[(p - n)v + (n + a - л)]

Now, we have that f e TSXi' Aa p a(i,v), then by previous corollary 4,

|an|<

p + a - л

p+(Ai+A2)(n-p)+b p+A2(n-p)+b

[(p - n)v + (n + a - л)]

which is

p+(Ai+A2)(n-p)+b p+A2(n-p)+b

[(p - n)v + (n + a - л)]

■\an\ < 1.

p + a - л

Since ^™ wn = 1, we see wn < 1, for each n > p. We can set that

Wn

p+(Ai+A2)(n-p)+b p+A2(n-p)+b

[(p - n)v + (n + a - л)]

Thus,

|an| = Wn

p + a - л

p + a - л

|an|.

p+(Ai+A2)(n-p)+b p+A2(n-p)+b

[(p - n)v + (n + a - л)]

The proof of theorem 3 is complete.

□

Corollary 5. The extreme points of TSm^b\2 p a(l,v) are the functions given by

fn(z) = zp+a -

p + a - л

p+(Ai+A2)(n-p)+b p+A2(n-p)+b

[(p - n)v + (n + a - л)]

-, (z £ U),

m

m

m

m

m

m

m

where n = 2,3,..., m,b G N0, A2 > \l > 0, p G N, a > 0,p > a, 0 < ¡i< p + a, v > 0.

Some other works related to subclasses of p-valent functions defined by other differential operators for different types of problems can be seen in [4; 5; 6; 8].

References

1. Al-Oboudi F.M. On univalent functions defined by a generalized Salagean operator. Int.. J. Math. Math. Sci., 2004, no 25-28, pp. 1429-1436.

2. Catas A. On a certain differential sandwich theorem a ssociated with a new generalized derivative operator. General Mathematics, 2009, vol. 17, no. 4, pp. 83-95.

3. Cho N.E. Certain classes of p-valent analytic functions. International Journal of Mathematics and Mathematical Sciences, 1993, vol. 16, pp. 319-328.

4. Choi J.H. On certain subclasses of multivalent functions associated with a family of linear operators. Advances in Pure Mathematics, 2011, vol. 1, pp. 228-234.

5. Darus M., Ibrahim R.W. Multivalent functions based on a linear operator. Miskolc Mathematical Notes, 2010, vol. 11, no. 1, pp. 43-52.

6. El-Ashwah R.M. Majorization Properties for Subclass of Analytic p-Valent Functions Defined by the Generalized Hypergeometric Function. Tamsui Oxford Journal of Information and Mathematical Sciences, 2012, vol. 28, no. 4, pp. 395-405.

7. Flett T.M. The dual of an inequality of Hardy and Littlewood and some related inequalities. Journal of Mathematical Analysis and Applications, 1972, vol. 38, pp. 746-765.

8. Ghanim F., Darus M. Some results of p-valent meromorphic functions defined by a linear operator. Far East J. Math. Sci., 2010, vol. 44, no. 2, pp. 155-165.

9. Kumar S., Taneja H., Ravichandran V. Classes multivalent functions defined by dziok-srivastava linear operator and multiplier transformations. Kyungpook Mathematical Journal, 2006, vol. 46, pp. 97-109.

10. MacGregor T.H. Functions whose derivative has a positive real part. Transactions of the American Mathematical ¡Society, 1962, vol. 104, pp. 532-537.

11. Salagean G.S. Subclasses of univalent functions. Proceedings of the Complex analysis-fifth Romanian-Finnish seminar, Part 1, Bucharest, 1013, 1983, pp. 362-372.

12. Umezawa T. Multivalently close-to-convex functions. Proceedings of the American Mathematical Society, 1957, vol. 8, pp. 869-874.

13. Uralegaddi B.A., Somanatha C. Certain classes of univalent functions. In. H. M. Srivastava and S. Owa (eds.), Current Topics in Analytic Function Theory, 1992, pp. 371-374.

Entisar El-Yagubi, Postgraduate, School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Bangi 43600, Selangor D. Ehsan, Malaysia, tel.: 0060123107493 (e-mail: entisar_e1980@yahoo.com)

Maslina Darus, Professor, School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Bangi, 43600, Selangor D. Ehsan, Malaysia, tel.: 0060133882683 (e-mail: maslina@ukm.edu.my)

Entisar El-Yagubi, Maslina Darus

О некоторых классах дробных р-валентных аналитических функций

Аннотация. Введены новые классы функций, обобщающие хорошо известные классы р-валентных функций, введенных ранее Т. Умезавой. Введены новые классы р-валентных функций, возникающих при применении формальных дифференциальных операторов и установлены достаточные условия принадлежности к ним. Также найдены определенные соотношения между этими классами.

Ключевые слова: аналитические функции; р-валентные функции; дифференциальный оператор; звездообразные функции

Entisar El-Yagubi, аспирант Школы математических наук (факультет науки и техники), Национальный университет Малайзии, Bangi 43600 Selangor D. Ehsan, Malaysia, tel.: 0060123107493 (e-mail: entisar_e1980@yahoo.com)

Maslina Darus, профессор Школы математических наук (факультет науки и техники), Национальный университет Малайзии, Bangi 43600 Selangor D. Ehsan, Malaysia, tel.: 0060133882683 (e-mail: maslina@ukm.edu.my)