SOME SUBORDINATION RESULTS FOR CERTAIN CLASS WITH COMPLEX ORDER DEFINED BY SALAGEAN TYPE Q-DIFFERENCE OPERATOR Текст научной статьи по специальности «Математика»

Vladikavkaz Mathematical Journal 2020, Volume 22, Issue 4, P. 7-15

YAK 517.53 + 517.54

DOI 10.46698/q5183-3412-9769-d

SOME SUBORDINATION RESULTS FOR CERTAIN CLASS WITH COMPLEX ORDER DEFINED BY SALAGEAN TYPE q-DIFFERENCE OPERATOR

M. K. Aouf1 and T. M. Seoudy23

1 Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt;

2 Department of Mathematics, Faculty of Science,

Fayoum University, Fayoum 63514, Egypt; 3 Department of Mathematics, Jamoum University College, Umm Al-Qura University, Makkah, Saudi Arabia E-mail: [email protected], [email protected], [email protected]

Abstract. The theory of the basic quantum calculus (that is, the basic q-calculus) plays important roles in many diverse areas of the engineering, physical and mathematical science. Making use of the basic definitions and concept details of the q-calculus, Govindaraj and Sivasubramanian [10] defined the Salagean type q-difference (q-derivative) operator. In this paper, we introduce a certain subclass of analytic functions with complex order in the open unit disk by applying the Salagean type q-derivative operator in conjunction with the familiar principle of subordination between analytic functions. Also, we derive some geometric properties such as sufficient condition and several subordination results for functions belonging to this subclass. The results presented here would provide extensions of those given in earlier works.

Key words: analytic function, subordinating factor sequence, hadamard product (or convolution), q-de-rivative operator, Salagean operator.

Mathematical Subject Classification (2010): 30C45, 30C50.

For citation: Aouf, M. K. and Seoudy, T. M. Some Subordination Results for Certain Class with Complex Order Defined by Salagean Type q-Difference Operator, Vladikavkaz Math. J., 2020, vol. 22, no. 4, pp. 7-15. DOI: 10.46698/q5183-3412-9769-d.

1. Introduction

Let A denote the class of functions of the form:

f (z) = z + £ ak zk, (1.1)

k=2

which are analytic in the open unit disc U = {z € C : |z| < 1}. We also denote by K the class of functions f € A that are convex in U. For two functions f and g, analytic in U, we say that f is subordinated to g in U, written f (z) x g (z), if there exists a Schwarz function w (z), which (by definition) is analytic in U with w (0) = 0 and |w (z)| < 1, such that f (z) = g (w (z)), z € U. Furthermore, if the function g is univalent in U, then (see [1, 2])

f (z) x g(z) & f (0)= g(0) and f (U) C g(U).

© 2020 Aouf, M. K. and Seoudy, T. M.

Given two functions f, g € A, where f is given by (1.1) and g is given by

ro

g(z) = z + £ bk zk, (1.2)

k=2

the Hadamard product (or convolution) f * g is defined by

ro

(f * g) (z) = z + £ akbkzk = (g * f) (z). k=2

For 0 < q < 1, the q-derivative of a function f € A is defined by (see [3-9])

nf () Jf(0)> z = 0,

I (q-1)* ' ' '

and D2f(z) = Dq(Dqf(z)). From (1.3), we have

Dqf (z) = l + 5>], afczk-1, (1.4)

k=2

where

1 - qk 1 , _ , , , fc-1

L J</ 1 - q

and

= 1+ q + q2 + ... + qk-1, (1.5)

f (qz) - f (z)

lim Dqf(z) = lim —---= f'(z),

(q - 1)z

for a function f which is differentiable in a given subset of C.

For f € A, Govindaraj and Sivasubramanian [10] defined the Salagean type q-difference operator as follows:

D0f (z) = f (z) Df (z)= zDqf (z) = z + 5>], akz'

œ

k

q

k=2

œ

)2f (z) = zDq (Df (z)) = z + ^ ([k]q)2 akzk,

k=2

Df (z) = zDq (Dn-1f (z)) , n € N = {1, 2, 3,...}

yq J W — ^q V^q

It is easily see that

Df (z) = z + ^([k]q)" akzk, n € N0 = N U {0}. (1.6)

k=2

We note that

œ

lim Df (z) = Dnf (z) = z + Y^ akzk, n € No.

q^1 k=2

The differential operator Dn was introduced and studied by Salagean [11] (see also Srivastava and Aouf [12]).

Let Gn (A, b, A, B) denote the subclass of A consisting of functions f (z) which satisfy

1 +

(1 — A) EllJ—1

+ ADq (Dqn/ (z)) - 1

1 + Az l + Bz

or satisfying

B - [B + (A - B) 6]

< 1,

b € C* = C\{0} ; 0 < A < 1; -1 < A < B < 1; 0 <B < 1; z € U. We note that:

(1.7)

(1.8)

(i) lim Gqn (A, b, A, B) = Gn (A, b, A, B) (see [13])

Dn / (z) ,

(1 - A)-+ A (Dnf (z)) - 1

1 + Az } 1 + Bz J '

(ii) lim Gqn (A, b, 1, -1) = Gn (A, b) (see [14])

= <^ / € A : R 1 +

Dn/ (z)

> 0 ;

(iii) GO (A,b, A, B)= Gq (A, b, A, B)

(1 - A) + XDqf (z) - 1

-<

1 + Az } 1 + Bz J '

1

b

(iv) Gqn (A, b, -1,1)= Gqn (A, b)

H / €

R 1 +

(1 - A)

Dn/ (z)

(v) Gqn (0,1 - a)= Gqn (a) = / €

: R

Dn/ (z)

+ ADq (Dn/ (z)) - 1

> a, 0 ^ a < 1

>0

Gqn (0,1 - a) = Rq1 (a) = {/ € A : R [Dq (D/ (z))] > a, 0 < a < 1}

1

b

z

z

(vi)

p (A, 1 - a, -1,1)= Gn

= / €

R

q (A,a)

f(1 - A)

Dn Dq

/ (z)

+ ADq (Dn/ (z))

>a

0 < a < 1;

z

(vii) Gqn ( A, e-ie (1 - a) cos 0, -1,1 ) = Gqn (A, a, 0)

= ^ / € A : R e'

Ad

(1 - A)

Dn/ (z)

+ ADq (Dn/ (z))

> a cos 0

n

\0\ < -, 0 < a < 1.

z

2. Main Result

To prove our main result we need the following definition and lemmas. Definition 1 (Subordinating Factor Sequence [15]). A sequence {bkof complex numbers is said to be a subordinating factor sequence if, whenever f (z) of the form (1.1) is analytic, univalent and convex in U, we have the subordination given by

E flkbkzk x f (z), z € U, ai = 1. k=1

Lemma 1 [15]. The sequence {bkis a subordinating factor sequence if and only if

1 + 2 E bkzkl > 0, z € U.

k=1

Now, we prove the following lemma which gives a sufficient condition for functions to belong to the class Gqn (A, b, A, B).

Lemma 2. Let the function f which is defined by (1.1) satisfy the following condition:

œ

E (1 + B ) {1 + A ([k]q - 1) } ([k] q)n |ak | < (B - A) |b|

(2.1)

k=2

then f € Gqn (A,b,A,B).

< Suppose that the inequality (2.1) holds. Then we have for z € U,

(1 - A)

Df (z)

+ ADq (Df (z)) - 1

B

+ ADq (Df (z))

- B - (A - B) b

akz

1

akz

k1

E{1 + A ([fc]q - 1 k=2

œ

(B - A) b + BE {1 + A k=2

œ

^E {1 + A ([k]q - 1)}([k]q)" |ak||z|k k=2

œ ^

_ , (B - A) |b| - B E {1 + A ([k]q - 1) } ([k] J™ |ak| |z|k—1

k=2

œ

^ E (1 + B) {1 + A ([k]q - 1) } ([k] J™ |ak| zk-1 - (B - A) |b| < 0,

k=2

which shows that f (z) belongs to the class Gqn (A, b, A, B). >

Let (A, b, A, B) denote the class of functions f (z) € A whose coefficients satisfy the condition (2.1). We note that Gqn* (A, b, A, B) C Gqn (A, b, A, B). Also, let Gq°* (A,b, A, B) = Gq* (a, b, A, B), Gqn* (a,b, -1,1) = Gqn* (A,b), Gqn* (A, 1- a,-1,1) = Gqra*(A,a:), Gqra* (A, (1 — a) cos —1, l) = Gqra* (A, b, 9) (|0| < f, 0 < a < l).

z

In this paper we prove several subordination relationships involving the functional class Gq7-* (A, b, A, B) employing the technique used earlier by Attiya [16] and Srivastava and Attiya [17] (see also [13, 14, 18-22]).

Theorem 1. Let the function f defined by (1.1) be in the class GG™* (A, b, A, B) and g € K. Then

(1 + B)(1 +*,)(!+,)• V9)(zH9(z) (M)

_2 [(1 + B) (1 + Aq) (1 + q)n + (B - A) |b

cnif(z)], q + B)(l + Ag)(l + gr + (B-A)|6| (1 + B) (1 + Xq) (1 + q)n '

The constant factor 111 subordination result (2.2) cannot be

replaced by a larger one.

oo

< Let f € Gqn* (A, b, A, B) and let g (z) = z + £ ckzk € K. Then we have

k=2

(1 + B)(1 + A,)(1 + ,

2[(1 + B) (1 + Aq) (1 + q)n + (B - A) (1 + B)(1 + Aq) (1 + q)n

oo

z

+ E akCk zk

(2.4)

2[(1 + B)(1 + Aq) (1 + q)» + (B - A) |b|], y k=2

Thus, by Definition 1, the subordination result (2.2) will hold true if the sequence

(1 + B)(1 + Aq) (1 + q)n

(1 + B) (1 + Xq) (1 + q)n + (B - A) \b\ak)k=l (2'5)

is a subordinating factor sequence, with a1 = 1. In view of Lemma 1, this is equivalent to the following inequality:

Since

tf(k) = {1 + A ([k]q - 1) } ([k] J" , k ^ 2, 0 < A < 1, 0 < q < 1, n € No, is an increasing function of k (k ^ 2), when |z| = r < 1, we have

tt Jl iV (l + B)(l + Ag)(l + gr fcl 1 ¿i(l + 5)(l + Ai)(l + ir + (B-A)|6|afcZ J J-, | (l + 2?)(l + Ag)(l + g)"

1 (1 + B) (1 + Xq) (1 + q)n + (B — A) \b\

oo

(1 + B)£(1 + Aq) (1 + q)n

.__fc=2_ k

+ (l + B)(l + \q)(l+qr + (B-A)\brZ

(l + B)(1 + A„)(l+,r g(l+B){l + A([tla-l)}([tl,)"|„,|

> 1----—-—---r--r

(1 + B) (1 + Xq) (1 + q)n + (B — A) \b\ (1 + B) (1 + Xq) (1 + q)n + (B - A) \b\

(l + B)(l + Ag)(l + g)w (B-A)|6|

(1 + B) (1 + Aq) (1 + q)ra + (B - A) |b| (1 + B) (1 + Aq) (1 + q)ra + (B - A) |b|

= 1 - r > 0, |z| = r < 1,

where we have also made use of assertion (2.1) of Lemma 2. Thus (2.6) holds true in U and also the subordination result (2.2) asserted by Theorem 1. The inequality (2.3) follows from (2.2) by taking the convex function g(z) = = z + 2zk- To prove the sharpness of the

constant 2[(i+B)(i+\g)ti+lr~+(b-A)|&|] > we consider the function /0 (z) € G™* (A, b, A, B) given by

{B-A)\b\

/o-"" {i + B){i + \q){i + q)nZ ■ (2'7)

Thus from (2.2), we have

(l + B)(l + A,)(l + ,r zeU. (2.8)

2[(1 + B)(1 + Aq) (1 + q)n + (B - A) |b|]'uv 7 1 - z' Moreover, it can easily be verified for the function f0 (z) given by (2.7) that

S?S r Ul(l + B) (1 + \q) (1 + „)" + (B - A) M f° Wj } = ~ 2 <2'9>

This shows that the constant 2[(l+Jxi+ig)(i+gj"V(b-A)|&|] P°ssible, this completes

the proof of Theorem 1. >

Putting n = 0 in Theorem 1, we have

Corollary 1. Let the function f defined by (1.1) be in the class G* (A, b, A, B) and g € K. Then

(1 + 5)(1 + A(?) 1 (/*</)(*) ^<7(*), ze U, (2.10)

,2 [(1 + B)(1 + Aq) + (B - A) |b|]

m{f(z)}>-(1 + Bni + Xq) + (B-A) 161 (2ii)

(1 + B)(l + Aq) *

The constant factor 111 subordination result (2.10) cannot be rep-

laced by a larger one.

Putting A = -1 and B = 1 in Theorem 1, we have

Corollary 2. Let the function f defined by (1.1) be in the class Gn (A, b) and g € K. Then

(1 + Aq) (1 + q)n

,2 [(1 + Aq) (1 + qf + |b|] and

(f * g) (z) x g (z), z € U, (2.12)

The constant factor 2[(i+aq)(:l-+g)"+1b\] 113 subordination result (2.12) cannot be replaced by a larger one.

Putting b = 1 — a(0 ^ a < 1), A = —1 and B = 1 in Theorem 1, we have

Corollary 3. Let the function f defined by (1.1) be in the class G™* (A, a) (0 ^ a < 1) and g € K. Then

(1 + A<?)(1 + yy i(f*9)(z)*g(z), ze U, (2.14)

2 [(1 + Aq) (1 + q)n + 1 - a]

The constant factor 2[(1+aI?)(i+<?!"+ 1 -«] 113 subordination result (2.14) cannot be replaced by a larger one.

Putting b = e~id (1 - ol) cos 0 (|0| < f; 0 ^ a < l) , A = -1 and B = 1 in Theorem 1, we have

Corollary 4. Let the function / defined by (1.1) be in the class G™* (A,a,9) (l^l < 0 ^ a < 1) and g € K. Then

(1 + A(?)(1 + <?r ze U, (2.16)

,2[(1 + Aq) (1 + q)n + (1 - a) cos 0]

and

The constant factor 2[(i+Agxt+g)"+(i-«)cose] 111 subordination result (2.14) cannot be replaced by a larger one.

Remark 1. Taking A = -1, B = 1 and letting q — 1- in Theorem 1, we get the result obtained by Aouf [14, Theorem 1].

Remark 2. Replacing A by -A, B by —B and letting q — 1- in Theorem 1, we obtain the result get by Sivasubramanian et al. [13, Theorem 2.2].

Remark 3. Putting n = A = 0, b = 1 — a (0 ^ a < 1) and letting q — 1- in Corollary 2, we get the result obtained by Aouf [14, Corollary 3].

Remark 4. Putting n = 0, A = 1, b = 1—a (0 ^ a < 1) and letting q — 1- in Corollary 2, we get the result obtained by Aouf [14, Corollary 4].

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Received April 1, 2020 Mghamed K. Aouf

Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt, Professor

E-mail: [email protected]

https://orcid.org/0000-0001-9398-4042;

Tamer M. Seoudy

Department of Mathematics, Faculty of Science, Fayoum University, Fayoum 63514, Egypt, Associate Professor;

Department of Mathematics, Jamoum University College, Umm Al-Qura University, Makkah, Saudi Arabia, Associate Professor

E-mail: [email protected], [email protected]

https://orcid.org/0000-0001-6427-6960

Владикавказский математический журнал 2020, Том 22, Выпуск 4, С. 7-15

НЕКОТОРЫЕ РЕЗУЛЬТАТЫ О ПОДЧИНЕНИИ ДЛЯ ОДНОГО ФУНКЦИОНАЛЬНОГО КЛАССА, ОПРЕДЕЛЯЕМОГО ^-РАЗНОСТНЫМ ОПЕРАТОРОМ ТИПА САЛАГИНА

Ауф М. К.1, Сеуди Т. М.2>3

1 Университет Мансура, Мансура 35516, Египет; 2 Университет Файюма, Файюм 63514, Египет; 3 Университет Умм-Аль-Кура, Мекка, Саудовская Аравия E-mail: [email protected], [email protected], [email protected]

3

1

2

Аннотация. Теория базового квантового исчисления (то есть базового q-исчисления) играет важную роль в различных областях знания, инженерии, физико-математических науках. Используя основные определения и некоторые детали q-исчисления, Говиндарадж и Сивасубраманиан [10] определили q-разностный (q-производный) оператор типа Салагина. В этой статье мы вводим определенный подкласс аналитических функций со сложным порядком в открытом единичном круге, применяя q-производный оператор типа Салагина в сочетании с известным принципом подчинения между аналитическими функциями. Кроме того, мы выводим некоторые геометрические свойства и несколько результатов о подчинении для функций, принадлежащих этому подклассу. Представленные здесь результаты расширяют результаты, представленные в более ранних работах.

Ключевые слова: аналитическая функция, подчиняющая фактор последовательность, произведение Адамара (или конволюция), q-производный оператор, оператор Салагина. Mathematical Subject Classification (2010): 30C45, 30C50.

Образец цитирования: Aouf M. K. and Seoudy T. M. Some Subordination Results for Certain Class with Complex Order Defined by Salagean Type q-Difference Operator // Владикавк. мат. журн.—2020.— Т. 22, № 4.—C. 7-15 (in English). DOI: 10.46698/q5183-3412-9769-d.