Certain subclasses of analytic functions defined by a new general linear operator Текст научной статьи по специальности «Математика»

Онлайн-доступ к журналу: http: / / mathizv.isu.ru

Серия «Математика»

2018. Т. 24. С. 24-36

УДК 517.547 MSG 30С45, 30С80

DOI https://doi.org/10.26516/1997-7670.2018.24.24

Certain subclasses of analytic functions defined by a new general linear operator

Abdul Rahman Salman Juma

University of Anbar, Baghdad, Iraq

Maslina Darus

Universiti Kebangsaan Malaysia, Bangi Selangor, Malaysia

Abstract. Hypergeometric functions are of special interests among the complex analysts especially in looking at the properties and criteria of univalent. Hypergeometric functions have been around since 1900's and have special applications according to their own needs. Recently, we had an opportunity to study on q-hypergeometric functions and quite interesting to see the behavior of the functions in the complex plane. There are many different versions by addition of parameters and choosing suitable variables in order to impose new set of q-hypergeometric functions. The aim of this paper is to study and introduce a new convolution operator of q-hypergeometric typed. Further, we consider certain subclasses of starlike functions of complex order. We derive some geometric properties like, coefficient bounds, distortion results, extreme points and the Fekete-Szego inequality for these subclasses.

Keywords: analytic functions, univalent functions, starlike functions, linear operator, Fekete-Szego problem

Recently, Darus and others in [1] and [2] have used the ^-hypergeometric functions in studying certain families of analytic functions in the open unit disk. The q-hypergeometric functions are the generalized form of the classical hypergeometric function. Then by letting the limit q —> 1, it will return to the classical hypergeometric function. The notion of hypergeometric functions have been used and introduced by many great mathematicians started by Euler in (1748), Gauss (1813) and Cauchy (1852) and after

1. Introduction

that, Heine(1846) converted a simple notation lim^i = a a systematic theory of hypergeometric functions parallel to the theory of Gauss hypergeometric function.

Here, we can say that many of the results for classical hypergeometric functions can be generalized to g-hypergeometric functions.

In this work, we shall introduce a new subclasses of univalent functions involving new operator Cqa(ai, bj) which generalizes many well-known operators and derived some geometric properties for this new subclasses. Denote A the class of functions / of the form

oo

f(z) = z + Y/anzn (1.1)

n= 2

which are analytic and univalent in the open unit disk U := {z € C : \z\ < 1}. A function / € A is said to be starlike of complex order if the following condition (see[3]) is satisfied:

1 zf'(z)

+ ¿(-j^y - 1)} > /?, (0 < /? < 1 and de C\{0}) (1.2)

For complex parameters a\,....., at and bi,..., br(b3e C\{0,-1,-2,-3,....},

j = 1, ...r, \q\ < 1), the q-hypergeometric

oo

q, (ai,g)ra----(at,g)ra za ^ ^

1 T ^0i(l'(l)n(bi,q)n--(br,q)n

(t = r + 1; t, r € N0 = {0,1, 2,...}; z & U). The q-shifted factorial is defined by

(a,q)o = l and (a, q)n = (1 — a)(l — aq)(l — aq2)....(l — aqa~l), n € N,

where a any complex number and in terms of the Gamma function

, _Tq{a + n){l-qT

1

Tq{a)

such that Tg(y) = {q'q)^q)q^1 < < 1. We note that and by using ratio test, the series (1.3) converges absolutely in open unit disk W, \q\ < 1 and t = r + 1. Now,if t = 2 and r = 1, then we have the following

^ (q,q)n(bi,q)n is the q-Gauss hypergeometric function [4].

Recently, Mohammed and Darus [1] defined the the following: l(ai]bj-,q)f : A A :

Z(,ai]bj]q)f{z) = z+y ------------cnz . (1.4)

Q)n-i(bi,q)n-i....{br,q)n-i

The Srivastava-Attiya operator JSA : A —> A is defined in [10] as:

oo /11 \ s

Js,af(z) = z + £ J c^, (1.5)

where z eW, a € C\{0, —1, —2, —3,...}, s € C and f € A). This linear operator JSA can be written as

Ja,af(z) = G.%a(z) * f(z) = (1 + a)s^(z, s, a) - a~s) * f(z),

by using the Hadamard product (convolution). Here,

OO ,n

z"

n=0

is the well-known Hurwitz -Lerch zeta function(see [5], [6]). It is also an important function of Analytic Number Theory such the De Jonquiere function:

OO r

Z

Lis(z) = = zHz,s, 1), (Re(s) > 1 if И = 1).

n=о [-n)

We define the linear operator £д'а(ец, bj)(j) : A —> A as follows:

fs,a( h.\t(~\ {ai,q)n-i....{at,q)n-i (1 + aV ra

q Z + ¿(9>9)n_1(6i>9)n_i....(6r>9)n_i [n + aj CnZ ■

(1.6)

(z eU,ae C\{0, -l, -2, -3,...}, seC, сц, bj e C, bj e С \ {0, -l, -2,...}, \q\ < 1 and t = r + 1).

It should be noted that the linear operator (1.6) generalised many operators studied by several earlier authors as follows: 1- If s=0, then

C°qa(at,b3)f(z)=Mq(at,b3),

where Mq(ai,bj) is the linear operator introduced by Mohammed and Darus [1].

For q —>• 1,ец = qai,bj = q^, where сц, (3j € С and fij ф 0 (г = 1, 2,..., t and j = 1,2,.., r), we have the following operators: 2- If t = 2, r = 1, a\ = A + 1, «2 = A + 1, /?i = v + 1, then we obtain the

operator considered by Prajapat and Bulboaca [7].

3- If t = 2, r = 1, ai = \,d2 = l,fli = v + 1, then we have the operator considered by Noor and Bukhari [8].

4- If s = 0, a = 0, t = 2, r = 1, ai = A, «2 = 1, (3\ = v + 1, then we obtain the Choi-Saigo-Srivastava operator [9].

5- If t = 2, r = 1, ai = /3i, CK2 = 1, then we obtain the Srivastava-Attiya operator [10].

6- If s = —x,t = 2,r = l,a\ = Pi,a2 = 1, then we obtain the Cho and Srivastava operator [11].

7- If s = — k(k € IN), a = 0, t = 2, r = 1, a\ = X = [3\, «2 = 1, then we obtain the Salagean operator [12].

8- If s = 1, t = 2, r = 1, ai = /3i, CK2 = l,a > —1, then we obtain the Bernardi operator [13].

9- If s = 0, a = 0, t = 2, r = 1, ai = X, «2 = 1, = v, then we obtain the Carlson-Shaffer operator [14].

10- If s = 0, t = r + 1, then we obtain the Dziok-Srivastava Operator [15]. Some of these operators contained some other operators (see for example [16; 17])

Definition 1. A function f eW is said to be in the classSsq'a (d, (3) if the following condition holds:

pr ii | i (zcrMMz) \1

(d € C\{0} and 0 < /3 < 1)

Motivated by the work given by Srivastava and Gaboury [18], we investigate some geometric properties like coefficients estimate, distortion bounds, extreme points and the Fekete-Szegô problem for the current function class.

2. Coefficient Estimate

Theorem 1. For 0 < /3 < 1, d € C \ {0} and if f{z) € A satisfies the following

(ïït!)' * <">

then f e Sq'a(d,f3)

Proof Suppose that (2.1) holds. Then if

z{Cqa{ai,bj)f{z))' _ ^ 1 p _ ^^ _ ^

TT( \ _ dCqa(ai,bi)i(Z) d_

Z(£q'a(ai,bj)f(z))' _ p _ -Q ^

dCsq'a(ai,bj)f(z) d

Therefore its enough to show that \H(z)\ < l,that is

z(Csq'a(ai, dCq'a{ai bj)í(z)y bj)f(z) "(I + ß~ 1)-1

z(£q'a(ai, dCsq'a{ai bj)f(z))' bj)f(z) "(I + ß~ 1) + 1

= ßdz + En=2 J^^tia^ ßd)CnZ-

(2 - ß)dz - ЕГ=2 (s)e +1 - '

thus by using (2.1) we have

\H(z)\ <

£ (a) Vi-zw^r <-^--—--<

n= 2 4 '

№+£ ^fete^fefc (s)e tn-i-ß\d\)cn <-—--—--< i,

(2 -ßM - £ j^í^tía^ ii=2 4 7

and the proof is complete. □

3. Distortion Bounds

Theorem 2. For 0 < ß < 1, d € С \ {0} and let f(z) of the form (1.1) be in the classSq'a(d, ß). Then

r (1 - ß)\d\2r2 y^ (Q, g)n-i(fti, g)n-i--(ftr, g)n-i / n+_a\s <

< Ш\ <

(1 - ß)\d\2 2ул (q,q)n-i(bi,q)n-i--(br,q)n-i ín + a V

(|г|=г<1) (3.1)

(1 — P)\d\2 ^ n(q, q)n-i(b\, q)n-\....(br, q)n-i (n + a" J)P(J\ r .....

Re^ ^2(ai,q)u-i....(at,q)u-i(n-I + \1 + aJ

<l/'(*)l <

< 1| (1n{q,q)n-i{bi,q)n-i....{br,q)n-i in + a Re(d) "¿2{ai,q)n_l....{at,q)n.l{n- 1 + ¿^jf) U + a

(N=r<l) (3.2)

Proof. Let f(z) be of the form (1.1). Then by using Theorem 2.1, we have

oo

\f(z)\>\z\-J2K\\zn\>

n=2

> ^ (l~/?)M|2r2y^ (g,g)n-i(fti,g)n-i--(ftr,g)n-i | /n+aV ,

"" " ¿(a1>i)„-i....(at>i)„_i(n-l + ii^)lVl+a>'

and

oo

l/(*)l< W + £№nl<

n=2

(l—(3)\d\2 (g,g)ra_i(6i,g)ra_i....(6i.,g)ra_i +

< r : jy

a/

(\z\=r< 1)

Also from(l.l), we have

\f'(z)\ > 1 - Y^n\cn\\zn-l\

>

n=2

oo

^e(d) r^2(ai>9)n_1....(at>9)n_1(„-l + ii^) WW

and

l/'WI^i + EnWi^-1!

<

n=2

(1 — /?)|d|2 ^ n(g,g)ra_i(6i,g^-i.-^ftr-,g)ra_i (n + a

(N=r<l).

Therefore, the results (3.1) and (3.2) are obtained. The proof is complete.

□

4. Extreme Points

In this section we take the subclass Sg'a(d, (3) of the class Sq'a(d, (3) consisting of all the functions f(z) € A of the form (1.1) and satisfy (2.1).The following theorem determine the extreme points of the subclass Sg'a(d, (3)

Theorem 3. Let

and

fn(z) =

fi(z)=z

= z +

\{q,q)n-i(bi,g)n-i....{br,q)n-i\ (i-ß\d\) {n- ß\d\) I(ai, q)n-i~•• (at, q)n-i\

(4.1)

n + a 1 + a/

(Ml). (4.2)

Then f € Sq'a(d,(3) if and only if

oo / oo \

f(z) = r]nfn(z) I rin > 0; rin = 1 J

n= 1 \ n= 1 /

Proof Let / € Sq'a(d,(3). Then in virtue of (2.1), we can set

(4.3)

Vn

= (n — f3\d\)

(ai,q)n-i....(at,q)n-i

1 + a

|Cn|,

(1 - ß\d\)(q, q)n-i(bi, q)n-i....(br, q)n-i \n + ay

(n ф 1). (4.4)

which give our result (4.3). Conversely, let

f(z) = £>/»(*)

n= 1

n= 1

\(q,q)n-i(bi,q)n-i....(br,q)n-i\(1 - ß\d\) {n- ß\d\) \ {ai, q)n-i....{at, q)n-i\

n + a 1 + a

Then

f>-ß\d\), i"1'!!"-1;-^':!"-' (l+a ^ {q, q)n-i{bi,q)n-i--{br, q)n-1

n + a

■Vn

\(q,q)n-i(bi,q)n-i....(br,q)n-i\(1 - f3\d\)

{n - (3\d\) \ {ai, q)n-i....{at, q)n-i\

n + a 1 + a

1 — /5|rf| ^ ^ = (i _ /3|d|)(l — 77!).

n=2

Therefore, we have / € Sq'a(d,(3).

□

5. The Fekete-Szego Inequality

Let Sg'a((f)) be a a class consisting of all the functions f(z) € A of the form (1.1) and satisfies the following

1 /zC'q'aM)f(z) \ 1 + Az

1 +A V^i + Bz-™' (5"1}

(z eM, a € C\{0, -1, -2, -3,...}, sec, Oi, bj € C, bj € C \ {0, -1, -2,...}, \q\ < 1 and t = r + l,d € C\{0}), where denotes subordination ,~1<B<A<1.

In this section, we shall find the upper bounds of the Fekete-Szego functional for the classSg'a((f)). We need the following lemma due to Ma and Minda [19] to prove our theorem involving the Fekete-Szego inequality

Lemma 1. Ifp(z) = 1 + d\z + d2z2 +.... is an analytic function inU with positive real part, the for any complex number /x,

\d2 ~ < 2max{l, |2/x - 1|}. (5.2)

The result is sharp for functions given by

p{z) = ^ ^ z and p{z) = ^ ^ z

1-z v ' 1 - zi

Theorem 4. Let <f>(z) = 1 + B\z + B2z2 + ...,and f(z) given by (1.1) belongs to &g'a((f)). Then

|c3 — jLtc|| <2max{l,\— - 1|}. (5.3)

Proof. Let f(z) € Sg'a((f)). Then by definition of subordination, there exists a Schwarz functionty(z) withw(O) = 0 and \ w(z)\ < 1 , analytic in the open unit disk such that

Define

Pi(z) := ? = l + dlZ + d2z2 + ...

1 — w(z)

it is clear that Re(p\(z)) > 0 and pi(0) = 1. Let

p(z) :=l + l(-l)=l + hlZ + h2z2 +

d \ £q' (a,i, bj)f(z) Therefore, in view of the above equations, we have

,pi(z) - 1

p(z) = ф(

and since

Pi(z) - 1 = 1 Pi(z) +1 2

thus

Pi(z) + l'

d2 d3 diz + (d2 - )z2 + (d3 + -j - did2)z3 +

\B1(d2-^d2) + ^B2c2

(5.5)

(5.6)

z2 + ...

By comparing the coefficients for z we obtain

and

hi = ^Bidi

h2 = ^B1(d2-^d2) + ^B2d2.

Then, with the help of (1.6), we get

zCsq (fl¿, bj)f(z) _ ]_ i g ( (аь g)i fi + a

£q (ai) bj)f(z)

+ Í2c3

\(q,q)i(bi,q)i....(br,q)i \2 + a (a1,q)2....(at,q)2 fl + a

-C2

(Q,Qh(bi,q)2....(br,q)2 + a (ai,q)i....(at,q)i

11 \ 2

1 + \ 2 .

Z +

+ 3c4

+ C2

(q,q)i(bi,q)i....(br,q)i \2 + a

(ai, q)3....(at, д)з fl + a

(q,q)3(bi,q)3---(br,q)3 + a (ai,q)i....(at,q)i fl + a

+

s\ 3

-3 ça

(q,q)i(bi,q)i--(br,q)i \2 + a (ai,q)2....(at,q)2 fl + a

(q,q)2(bi,q)2....(br,q)2 \3 + a/

(ai,q)i....(at,q)i fl + a

c2

(q,q)i(bi,q)i--(br,q)i \2 + a

z+

zá + ...

Therefore, from (5.6) we obtain

{ai,q)i....{at,q)i /1 + a

Vll\ = C2~-:—7--:-j--I -

{q,q)i{bi,q)i---{br,q)i \2 + a (a1,q)2....(at,q)2 /1 + a

bh2 = 2 c3

(q,q)2(bi,q)2....(br,q)2 \3 + a

{ai,q)i....{at,q)i /l + axsx2

- c2-

'(i, 9)1(61,9)i-(6r,i)i V2 + a then

dBidi (q, q)i(bi,q)i....(br,q)i /2 + a c2 = —

(ai,g)i....(at,g)i Vi + a

C3 =

= (*Bld2 + ^(p2B2(Q,Qh(h,q)i....(br,q)i /HaV_ _ V4 8 (a1,q)1....(at,q)i \l + aj

(q,qh(bi,q)2....(br,q)2 /3 + a

(ai,q)2....(at,q)2 Vi + a

Therefore, we obtain

c3 - ¡icl =

V4 8 (ai,g)i....(at,g)i \l + a/

(q,qh(b1,q)2....(br,q)2 /3 + a

(ai,q)2....(at,q)2 Vi + a ^ V 2 (ai(,g)i....(at,g)i Vx + a

/dBid2 {q,q)i{bi,q)i....{br,q)i /2 + ax 2

Then where and

2 dB\a2 , 2 \ c3 - /xc2 = —-— [d2 - dxT) ,

Z ±J\ U 2

(g,g)i(&i,g)i--(&r,g)i ^2 + a

0"! =

(ai,q)i....(at,q)i Vi + a (q,qh(bi,q)2....(br,q)2 /3 + a

(ai,q)2....(at,q)2 Vi + a

Now, using Lemma 5.1 we get

|c3 - /xcgl < 2max{l, |— - 1|}.

The result is sharp for the functions / given by

1 + z2 1 + z

P(Z) = i-9 >P(Z) =

1-z

□

Here, we can mention that this result generalizes many recently results which investigated in several earlier works. In fact, some other work related to g-analoque can also be seen in [20-22].

Conflict of interest: The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgement: The work here is supported by UKM grant: GUP-2017-064.

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Abdul Rahman Salman Juma, Department of Mathematics, University of Anbar, 55431 Baghdad, 55 Ramadi, Iraq, tel.: 421119, 424878 (e-mail: [email protected])

Maslina Darus, Professor of School of Mathematical Sciences (Faculty of Science and Technology), Universiti Kebangsaan Malaysia, Bangi 43600 Selangor D. Ehsan, Malaysia, tel.: 0060133882683 (e-mail: [email protected])

Received 27.11.2017

Подклассы аналитических функций, определяемые общим линейным оператором

Abdul Rahman Salman Juma

University of Anbar, Baghdad, Iraq

Maslina Darus

Universiti Kebangsaan Malaysia, Bangi Selangor, Malaysia

Аннотация. Гипергеометрические функции вызывают особый интерес в теории функций комплексных переменных, особенно при рассмотрении свойств и критериев однолистных функций. Гипергеометрические функции существуют с 1900-х годов и имеют специальные приложения в соответствии с их собственными потребностями. Недавно у нас была возможность изучить q-гипергеометрические функции и увидеть довольно интересное поведение функций в комплексной плоскости. Существует множество различных версий путем добавления параметров и выбора подходящих переменных, чтобы получить новый набор q-гипергеометрических функций. Целью настоящей работы является изучение и введение нового оператора свертки с q-гипергеометрической функцией. Рассмотрены некоторые подклассы звездообразных функций сложного порядка. Получены некоторые геометрические свойства, такие как оценки коэффициентов, теоремы искажений, экстремальные точки и неравенство Фекете - Сегё для этих подклассов.

Ключевые слова: аналитические функции, однолистные функции, звездообразные функции, линейный оператор, задача Фекете - Сегё.

Abdul Rahman Salman Juma, 55431 Baghdad, 55 Ramadi, Iraq, tel.: 421119, 424878 (e-mail: [email protected])

Maslina Darus, Bangi 43600 Selangor D. Ehsan, Malaysia, tel.: 0060133882683 (e-mail: [email protected])

Поступила в редакцию 27.11.2017