On certain subclassesof analytic functions with varying arguments of coefficients Текст научной статьи по специальности «Математика»

Серия «Математика»

2018. Т. 23. С. 80-95

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

ИЗВЕСТИЯ

Иркутского государственного ■университета

УДК 518.517 MSG 30С45

DOI https://doi.org/10.26516/1997-7670.2018.23.80

On Certain Subclasses of Analytic Functions with Varying Arguments of Coefficients

H. M. Zayed

Menofia University, Egypt

M. K. Aouf

Mansoura University, Egypt

Maslina Darus *

Universiti Kebangsaan Malaysia

Abstract. In this paper we introduce and study the class V1Zs:V(n, А, a) ol analytic functions with varying arguments of coefficients. We obtain coefficients inequalities, distortion theorems involving fractional calculus, radii of close to convexity, starlikeness and convexity and square root transformation for functions in the class V1Zg>rl(ji, A, a). Finally, integral convolution for functions in this class are considered.

Keywords: analytic functions, Hadamard product, fractional calculus operators, varying arguments of coefficients, square root transformation, integral convolution.

1. Introduction

Let An denote the class of functions of the form:

oo

f(z) = z + akzk (n € N = {1,2,...}), (1.1)

fc=ra+l

which are analytic in the open unit disc U = {z : z € C and < 1} and let Sn be the subclass of all functions in An, which are univalent in U. Let S*(a) and K,(a) denote the subclasses of starlike and convex functions of order a (0 < a < 1). We note that S*(0) = S* and /C(0) = /C, the subclasses

* This research was supported by UKM grant: GUP-2017-064

of starlike and convex functions (see, for example, Srivastava and Owa [8]). Also, let IZ(a) (1 < a < 2) denote the subclass of A\ satisfies the inequality

»{/'(*)}< a. (1.2)

The class lZ(a) (1 < a < 2) was studied by Uralegaddi et al. [9].

Many essentially equivalent definitions of fractional calculus (that is, fractional derivatives and fractional integrals) have been given in the literature (cf., e.g. [1], [3], [5] and [6]). We find it to be convenient to recall here the following definitions which were used recently by Owa [3] and by Srivastava and Owa [7].

Definition 1. The fractional integral of order 5 is defined, for a function №), h

D7Sf{z> = W)£i^ihdas>0h

where f(z) is an analytic function in a simply-connected region of the complex z—plane containing the origin and the multiplicity of (z — C)5-1 is removed by requiring log(2 — () to be real when z — ( > 0.

Definition 2. The fractional derivative of order 5 is defined, for a function

by

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

Definition 3. Under the hypotheses of Definition 2, the fractional derivative of order k + 5 is defined by

r]k

Dkz+Sf(z) = j^D&zf{z) (0 < 5 < 1; k € N0 = N U {0}).

In this paper, we define the following subclass of An as follows. Definition 4. A function f(z) € An is said to be in the class lZs(n, A, a) if

K jr(2 - 5)zs~1 [(1 - A)DsJ(z) + XzDsz+1f(z)] } < a

(l<a<2; 0 < A < 1; 0 < 5 < 1; a + 5 <2-, zeU). (1.3)

In [4], Silverman introduced and studied the univalent functions with varying arguments of coefficients, as follows:

Definition 5. [4] We say that a function f(z) of the form (1.1) is in the class V(9k) if f(z) € S (the class of analytic and univalent functions in U) and arg(afc) = 9k for all k (k > 2). Further, if there exists a real number rj such that

9k + (k - l)r? = it {mod 2vr), (1.4)

then f(z) is said to be in the class V(9k,rf). The union ofV(9k,r/) taken over all possible sequences {9k} and all possible real numbers rj is denoted by V.

Silverman [4] used the concept of varying arguments of the coefficients to introduce and study the class V*(a), which is a subclass of V consisting of starlike functions of order a.

For rj = 0, we obtain the class Tn consisting of functions f(z) with negative coefficients.

Using the concept of varying arguments of coefficients in univalent functions, we introduce the following subclass.

Definition 6. Let VlZ$yV(n, A, a) denote the subclass of V consisting of functions f(z) € lZs(n, A, a).

We note that:

(i) V7Z0iV(n, A, a) = VTZv(n, A, a) =

€ An : К

(1_Л)Ж + Л f{z)

(ii) V7Z0>v(n, 0, a) = VRv(n, a) = {/(z) еЛ,:»^

< a

< a

(iii) VK0,v(n, 1, a) = VKv(n, a) = {f(z) еЛ,:» f\z) < a} ;

(iv) VKo,o(n, 0, a) = R(n, a) = {/(г) eT„:» < a} ;

(v) VTZ0,0 (n, 1, a) = Щп, a) = {f(z) eTn:S f'(z) < a} ;

(vi) ]лг0)0(1,1, a) = 11(1, a) = 11(a) (see Uralegaddi et al. [10]).

2. Coefficient estimates for the class V1ZsiV(n, X, a)

Unless otherwise mentioned, we assume throughout this paper that l<a<2, 0 < A < 1, 0 < 5 < 1, a + 5 <2, k>n + 1, neNandzeU.

Theorem 1. Let the function f(z) be given by (1.1). Then f(z) € VlZsiV(n, A, a) if and only if

E

fc=ri+1

r(2 - S)T(k + 1) [1 + AÇfc - S - 1)] T(k + 1 - 5)

\ak\<a + XÖ-l. (2.1)

Proof. Assume that the condition (2.1) holds, then it is sufficient to show the inequality (1.3) holds. We find that

T(2 - 6)*5-1 [(1 - A)D&J{z) + XzDz+1 f(z)] - (1 - XS)

~ r(2 — ö)T(k + !)[! + A(fc — £ — 1)] ^ fc_!

/ j T^/7. , 1 C\ (l^Z

k=n-\-1

T(k + 1- S)

Since f(z) € V, then f(z) € V(0fc,r?) for some sequence {9k} and a real number r] such that

Ok + (k — 1 )rj = 7r (mod 2tt) .

Let z = retr], we have

T(2 - S)^-1 i(l - X)DsJ(z) + XzDsz+1f(z)} - (1 - X5)

fc=ri+1 oo

r(fc + 1 - 5)

£ r(2-a)r(fc + i)[i + A(fc-a-i)] ^^

fc=ri+l oo

r(fc + 1 - 5)

< ^ r(2 — 5)T(k + !)[! + A(fc — £ — !)] ^

fc=ri+1

< Ö + A5-1.

T(k + 1 - 5)

This shows that the function

$(*) = r(2 - 5)zs~1 [(1 - A)Dszf(z) + AzDsz+1f(z)

lies in a circle which is centered at w = 1 — XS with radius a + XS — 1, hence the inequality (1.3) holds. Conversely, assume that

k|T(2 -S)Î

or, equivalently

{oo

1-X5 + Y^

k=n+1

S-l

(1 — X)D6zf(z) + XzDz+1 f(z)

< a,

r(2 — ö)T(k + !)[! + A(fc — £ — !)] fc_! I

T(k + 1-Ö) k p

Letting 2 —> we obtain the required result and hence the proof is completed. □

Corollary 1. Let the function f(z) defined by (1.1) be in the class

VJlsyV(n,\,a).

Then

(a + X5-l)T(k + l-5) |flfc| " r(2 - 5)T(k + 1) [1 + A (k-5- 1)]'

The result is sharp for the function

f(z) z + (a + A£ — 1) r(fc + 1 — d k

+T(2-5)T(k + l)[l + X(k-5-l)} 1 j

3. Distortion theorems involving fractional calculus for the

class VlZs,v(n, A, a)

Theorem 2. Let the function f(z) defined by (1.1) be in the class VlZ$yV(n, A, a), then for ¡j, > 0 and we have

\D~4(z)\ <

and

\D~4(z)\ >

L(2 + /x)

L(2 + /x)

(a + A£ — 1) Г(2 + ¡j)Y(n + 2 — £) ' Г(2 - 5)T(n + 2 + /х) [1 + A(n- 5)] И

_ (a + X5 - 1) Г(2 + /x)L(n + 2 - 5) . ,n Г(2 - 5)T(n + 2 + fj,)[l + X(n- 5)}

The result is sharp for the function f(z) given by

f(z) = z , (a + A* - 1) r(2 + /x)r(n + 2-S) ^ +1 n> T(2-5)T(n + 2 + fx)[l + X(n-5)] '

(3-1)

г

(3.2) (3-3)

Proof. It is easy to see from Theorem 1 that

Г(2 - ¿)Г(п + 2) [1 + A(n - ¿)] (а + А5-1)Г(п + 2-5)

oo

i £

fc=ri+1

fc=ri+1

Г(2 - 5)T(k + 1) [1 + A (k-5- 1)] (a: + A£ — 1) T(k + 1 — <5)

lflfcl

Ы < 1.

Hence

<

fc=ri+1

(a + XS - 1) T(n + 2 - 5) Г(2 - 5)Г(п + 2) [1 + X(n - 5)]

Let

F(z) = r(2 + ß)z~ßD~ßf(z)

~ r(fc + l)r(2 + ß) k

' / J T^n. I -1 I fc

fc=ri+1

T(k + 1 + /x)

(3.4)

Since

then

0 < r(fc + l)r(2 + ß) < r(n + 2)r(2 + ß)

T(k + 1 + ß)

T(n + 2 + ß)

\F(z)\ <

~ r(fc + l)r(2 +ß) k

' / j -n/7. , -1 , \

fc=ri+l oo

r(fc + 1 + ß)

r(fc + 1 + ß)

< W +

fc=ri+l

r(n + 2)r(2 + ß)

T(n + 2 + ß)

Y1 iafci

ira+l

fc=ri+l

< | (a + \ö-l)T(2 + ß)T(n + 2-ö) +1

and

№)|>M-

r(2 - 6)T(n + 2 + ß)[l + \ (n- 5)]

(a + A<S - 1) r(2 + ß)T(n + 2-5) +1

r(2 - 5)T(n + 2 + fi) [1 + \(n - 5)]

which proves the inequalities (3.1) and (3.2). Since each of equalities in (3.1) and (3.2) is satisfied by the function f(z) given by (3.3), the proof is thus completed. □

Theorem 3. Let the function f(z) defined by (1.1) be in the class VlZsiV(n, A, a), then for 0 < /x < 1 and z £ U, we have

№f(z)\ <

and

M№\ >

T(2-ß)

T(2-ß)

1 +

1 -

(a + A£ — 1) T(n + 2 — £) r(2 - 5)r(n + 1) [1 + A(n - 5)]

(a + A£ — 1) r(n + 2 — £) r(2 - 5)r(n + 1) [1 + A(n - 5)]

The result is sharp for the function f(z) given by

f(z) = z , (« + A£ — 1) r(n + 2 — £) ^ +1 J{ ) ^ r(2 - 5)T(n + 1) [1 + \{n - S)\ '

(3.5)

(3.6)

(3.7)

Proof. It is easy to see from Theorem 1 that Г(2 - 5)Г(п + !)[! + A(n - 5)]

(а + А5-1)Г(п + 2-5)

k\(lk\

k=n-\-1

Hence

Let

Since

Then

and

<

£

fc=ri+1

Г(2 - 5)T(k + l)[l + X(k-S- 1)]

(a + A5-l)r(fc + l- 5)

Ы < l.

k\ak\

k=n-\-1

<

(a + XS - 1) T(n + 2 - 5) Г(2 - 5)Г(п + !)[! + A(n - 5)]

G(z) = Г(2-M)z"D?f(z)

= z

~ Г (к + 1)Г(2 —ц) yk

* / j T^/7. i 1 .Л fc "

fc=Tl+1

T(k + 1 -¡л)

№)| <

+ 1)Г(2 -/x) 1 < Г(Л + 1 - /X) '

~ Г(А: + 1)Г(2 — ¡j) k

I / J ТЛП. , 1 . Л

fc=ri+l oo

T(fc + 1 - /i)

ra+l

fc=ri+l oo

< N + ^ к\ak\ \z

k=n-\-1

(a + A£ — 1) Г(та + 2 — £) +1 " 11 + Г(2 - 5)T(n + !)[! + A(n - 5)] 11

\G(z)\ > | - J^r.^ + ^L \z\n+1

Г(2 - 5)T(n + !)[! + X(n - 5)]

(3.8)

which proves the inequalities (3.5) and (3.6). Since each of equalities in (3.5) and (3.6) is satisfied by the function f(z) given by (3.7), the proof is thus completed. □

Putting /j, = 0 in Theorem 3, we obtain the following corollary

Corollary 2. Let the function f(z) defined by (1.1) be in the class VlZsiV(n,X,a), then for z £ V, we have

(a + X5-l)T(n + 2-5)

\№\<\z\ +

,\n+l

and

!/(*)!> W-

r(2 - 5)r(n + 1) [1 + A(n - 5)] 1 1

(a + X5-l)T(n + 2-5) +1 T(2-S)T(n + l)[l + X(n-S)] The result is sharp for the function f(z) given by

= ~ I (« + A£ — 1) r(n + 2 — £) ion+ n+1 J{ ) ^ r(2 - 5)T(n + 1) [1 + X{n - 5)]

Theorem 4. Let the function f(z) defined by (1.1) be in the class VlZsiV(n, A, a), then for ¡j, > 0 and z £ D, we have

\Dlz-»f(z)\ <

<

T(2+/x)

+ (a + XS- 1) (n+l+/x)r(2+/x)r(n+2 - /x)

r(2 - 6)T(n + 2 + ß) [1 + A(n - 6)}

and

>max 0,

r(2 + n)

, _ , _ (a + XS - 1) (n + 1 - fi)T(2 + ¡j)T{n + 2-fi) n 1 ^ T(2-5)T(n + 2 + ti)[l + X(n-5)]

Proof. Differentiating both sides of (3.4), we have

r(2 + f(z) - /xT(2 + ¡i)z~^~lf(z)

£

fc=ri+1

T(k + l)r(2 + n) T(k + l+ß)

kakZ

fc-i

(3.9)

Hence, we obtain the required result from (3.1), (3.2), (3.8) and (3.9). This completes the proof of Theorem 4. □

4. Radii of close-to-convexity, star likeness and convexity

Theorem 5. Let the function f(z) defined by (1.1) be in the class VR$yV(n, A, a). Then f(z) is close-to-convex of order /3 (0 < /3 < 1) in \z\ < r\, where

(T(2 — 5)T(k + !)[! + A(fc — £ — 1)] 1 k>n+i \ (a + XS-l)T(k + l-S) V k )} '

The result is sharp, the extremal function given by (2.2).

Proof. We must show that

|f'(z) - l| < 1 - /3 for \z\ < n, where r\ is given by (4.1). Indeed we find from (1.1) that

\f'{z)-l\ < Y, kak\z

k=n-\-1

к-1

Thus if

E

f(z)-l\ <1-13, к

1-/3

\ak\\z\k~l <1.

(4.2)

fc=ri+1

But by using Theorem 1, (4.2) will be true if к

<T_(2-5)T(k + l)[l + \(k-5-l)]

Then

\z\ <

l-f3j'~' ~ (a +X5-l)T(k + l-5)

T(2 - 5)T(k + 1) [1 + X(k - 5 - 1)] /1 - f3\\ ^

(a + XS - 1) T(k + 1 - 5) V k This ends the proof. □

Theorem 6. Let the function f(z) defined by (1.1) be in the class VlZ$yV(n, X, a). Then f(z) is starlike of order (3 (0 < (3 < 1) in \z\ < f2, where

= ( T(2 — 5)T(k + !)[! + X(k — ^ — 1)] fl-(3 2 k>n+i\ (a +X5-l)T(k + l-5) \k-(3

The result is sharp, with the extremal function given by (2.2).

Proof. We must show that

zf'(z)

1

fc-1

(4-3)

f(z)

- 1

<1-/3 for \z\ < Г2,

where r2 is given by (4.3). Indeed we find from (1.1) that

i-V^I

ifc-i

zf'(z)

f(z)

-1

<

E (k-l)ak\z\

k=n-\-1

i- E ak\z

k=n-\-1

fc-1

Thus

zf'(z) _ 1

<1-/3,

if

k=n+1 V H7

But by using Theorem 1, (4.4) will be true if

k - ß\ fc_ i < T(2 - S)T(k + 1) [1 + A (k-S- 1)]

.1-/37 ~ (a + XS-l)T(k + l-S) Then

( T(2 — 5)T(k + !)[! + A(fc — £ — !)] " \ (a + XS-l)T(k + l-S) \k — /3 J )

This completes the proof of Theorem 6. □

Using similar arguments to those in the proof of the Theorem 6, we obtain the following corollary

Corollary 3. Let the function f(z) defined by (1.1) be in the class VlZ$yV(n, X, a). Then f(z) is convex of order /3 (0 < /3 < 1) in \z\ < where

f T(2 — 5)T(k) [1 + X(k — £ — 1)] fl-p\ï ^

r3 fc>n+i\ (a + X5-l)T(k + l-5) \k-(3 The result is sharp, with the extremal function given by (2.2).

5. Square root transformation for the class V1Z$)7?(n, A, a)

Definition 7. [2] If f(z) € S and h(z) = \/f(z2), then h(z) € S and h(z) = z + Y^k=n+1 c2k-iz2k~l (z £ U). The function h(z) is called a square root transformation.

Theorem 7. If f(z) € VTZStV(n, A, a), (a + XS - 1) T(n + 2 - 6) < T(2 -¿)r(n + 1) [1 + A(n — and h(z) be the square root transformation of f(z), then

rl (a + X6-l)T(n + 2-6) V T(2-S)T(n + l)[l + X(n-S)]

< \Kz)\ <

r4 L , (a + X5-l)T(n + 2-5) V tr(2-i)r(n + l)[l + A(n-i)] '

The result is sharp for the function

f(7) = , , (a + XS-l)T(n + 2-6) w 2n J{ ) ^ r(2 - 5)T(n + 1) [1 + \{n - S)\

Proof. In view of Corollary 2, we have

2 _ (a + AS - 1) r(n + 2 — 5) 2(n+l) r(2 - 5)r(n + 1) [1 + A(n - 5)]

< |/(*2)| <

2 (a + A^-l)r(n + 2-g) 2(n+1) r(2 - 5)T(n + 1) [1 + A(n - 5)]

We find that

1ВД| = V\IW\

< rJ 1 +

(a + AS - 1) Г(п + 2 - S) ^

Г(2 - 6)T(n + 1) [1 + A(n - 5)] and

,w м / (a + \6 -l)T(n + 2-6)

- rV 1 - Г(2 — ¿)Г(п + 1) [1 + A(n - ¿)]

This completes the proof of Theorem 7. □

6. Integral convolution for the class VTZs)J?(n, A, a) Let fj(z) (j = 1,2) be defined by

oo

fj(z) = z+ E aktjzk, (6.1)

fc=ri+1

then, the integral convolution of fi(z) and /2(2) is defined by

oo

(/l®/2)(*)=2+ E aj^AZk = {f2®fl){z).

k=n-\-1

Theorem 8. Xei fj(z) (j = 1,2) defined by (6.1) be in the class VlZsyV(n,X,a).

Then (/i © f2)(z) € VJls,v(n, A, (), where

(a + AS - l)2T(n + 2 - S)

( = 1-X6 +

(n + 1) r(2 - S)T(n + 2) [1 + X(n - 5)]' The result is sharp for the functions fj(z) (j = 1,2) given by

f (z) z + + + c^-n+i a 1 2)

h[z) ~Z + T(2- 5)T(n + 2) [1 + X(n -5)f Z '

Proof We need to find the largest ( such that

^ r(2 - S)T(fc + 1) [1 + X(k - 6 - 1)] \akA\|afc,2| ^ (( + AS - l)r(fc + 1 - S) k ~ '

fc=ri+1

Since fj(z) € VlZ$yV(n, X, a) (j = 1,2), we readily see that

v- r(2 - ¿)r(fc + 1) [1 + A(fc - i - 1)]

(o + «-l)r(* + l-i) l^'l-1'

and

k=n+l (° + XS-mk + l-S) By the Cauchy Schwarz inequality we have

~ r(2 — S)T(fc + !)[! + A(fc — S — 1)]

^(2 — 5)T(k + !)[! + A(fc — S — 1)] / -(a + AS — l)r(fc + 1 — S)-

Thus it is sufficient to show that

T(2 - S)T(fc + 1) [1 + X(k - 5 - 1)] |aM||afc;2|

<

(( + AS - l)r(fc + 1 - S) k

T(2 - 5)T(k + 1) [1 + X{k - S - 1)] r.----

-(a + xs-i)r(k + i-S)-VI^ lafc'2'

(a + AS - 1)T(A; + 1 - S)

or, equilvalently, that

f.---; , / ( + AS — 1

VWW^ia + Ai-1,

Hence, in light of the inequality (6.3), it is sufficient to prove that (a + AS - l)r(fc + 1 - S) ( ( + AS — 1

^ rC

T(2 - 5)T(k + 1) [1 + X(k - S - 1)] ~ V" + ^ - 1

It follows from (6.4) that

(a + X5 - l)2T(k + 1 - 5)

(>1-XÔ +

kT(2 - S)T(k + l)[l + X(k-S- 1)] ' Now defining the function H(k) by

(a + X5 - l)2T(k + 1-5)

H (к) = 1 -X5 +

kT(2 - ô)T(k + 1) [1 + X(k - S - 1)]

we see that H(k) is an decreasing function of k. Therefore, we conclude that

C>*<» + 1) = 1-M + C + «-l)ar(» + 2 -i)

(n + 1) T(2 - 5)r(n + 2) [1 + X(n - 5)] '

which evidently completes the proof of Theorem 8. □

Using similar arguments to those in the proof of Theorem 8, we obtain the following theorem.

Theorem 9. Let fi(z) defined by (6.1) be in the class VlZ$yV(n, A, a) and /2(2) defined by (6.1) be in the class VlZ$yV(n, A, 7). Then (/1 ©/2X2) € VlZ$yV(n, A, £), where

ç = 1_XÔ+ (a + X0-l)(j + X0-l)T(n + 2-0)

(n + 1) r(2 - 5)r(n + 2) [1 + A(n - 5)]'

The result is sharp for the functions fj(z) (j = 1,2) given by

ff?1_?+ (a + X6-l)T(n + 2-6) jen+ n+1 Il{' j ~ + T(2 - S)T(n + 2) [1 + A(n - 5)}

and

f (?)-?+ (7 + Xd-l)F(n + 2-6) a +1 M j + T(2 - ¿)T(n + 2) [1 + A(n - ¿)]

Theorem 10. Let fj(z) (j = 1,2) defined by (6.1) be in the class VTZstV(n, X, a). Then the function

Hz) = z+ E --1-z

fc=ri+1

л

belongs to the class VlZ$yV(n, A, %), where

2(a + XÔ - l)2T(n + 2 - Ô)

X = 1 " XÔ +

(n + 1) Г(2 - 5)Г(п + 2) [1 + A(n - 5)]

The result is sharp for the functions fj(z) (j = 1,2) given by (6.2). Proof. By using Theorem 1, we obtain

~ fT(2-£)r(fc + l)[l + A(fc-£-l)n2,„ ,2< (a + X5-l)T(k + l-5) J |flfc'11 "

f ^ T(2 — 5)T(k + !)[! + A(fc — 5 — 1)] l'

(a + A5 — l)T(k + 1 — 5) W) <1, (6-5)

and

~ fr(2-5)r(fc + l)[l + A(fc-5-l)nV 2 ¿^l (a + X5-mk + l-5) / |afc'2' ^

[ ^ T(2 — 5)T(k + !)[! + A(fc — 5 — 1)] \2

(a + A5 — l)T(k + 1 — 5) M) ^ M

It follows from (6.5) and (6.6) that

~ 1 fr(2-5)r(fc + l)[l + A(fc-5-l)n2^ 2 j^U (l-X5-a)nk + l-5) ) U^l + ^

Therefore, we need to find the largest % such that

r(2 - 5)T(k + 1) [1 + A(k-5- 1)]

12 21

<

k(x + X5- 1 )T(k + 1-5)

1 f T(2 - 5)T(k + 1) [1 + A (k-5- 1)] " 2

2 \ (a + X5 - 1 )T(k + 1-5)

X > 1 - xs +

that is

2(a + X5- l)2T(k + 1-5)

kT( 2 - 5)T(k + l)[l + X(k-S- 1)]' Now defining the function I(k) by

2(a + X5 - l)2T(k + 1 - 5)

I(k) = 1-X5 +

< 1.

kT( 2 - 5)T(k + l)[l + X(k-S- 1)]' we see that I(k) is an decreasing function of k. Therefore, we conclude that

(>I(n + l)-l-X5 +_2(q + A5 — l)2T(n + 2 — 5)_

which evidently ends the proof. □

Remark 1. For different choices of 5, r] and A, we will obtain new results for different choices mentioned in the introduction.

References

1. Aouf M.K. On fractional derivatives and fractional integrals of certain subclasses of starlike and convex functions. Math. Japon., 1990, vol. 35, no 5, pp. 831-837.

2. Duren P.L. Univalent Functions. Springer-Verlag, New York,1983.

3. Owa S. On the distortion theorems. I. Kyungpook Math. J., 1978, vol. 18, no 1, pp. 53-59.

4. Silverman H. Univalent functions with varying arguments, Houston J. Math., 1981, vol. 7, no 2, pp. 283-287.

5. Srivastava H.M., Aouf M.K. A certain fractional derivative operator and its applications to a new class of analytic and multivalent functions with negative coefficients. I and II, J. Math. Anal. Appl., 1992, vol. 171, pp. 1-13; ibid. 1995, 192, pp. 973-688.

6. Srivastava H.M., Aouf M.K. Some applications of fractional calculus operators to certain subclasses of prestarlike functions with negative coefficients. Computers Math. Appl, 1995, vol. 30, no 1, pp. 53-61. https://doi.org/10.1016/0898-1221(95)00067-9

7. Srivastava H.M., Owa S. An application of the fractional derivative, Math. Japon., 1984, vol. 29, pp. 383-389.

8. Srivastava H.M., Owa S. (eds.). Univalent Functions, Fractional Calculus and Their Applications. Halsted press (Ellis Horwood Limited Chichester). John Wiley and Sons, New York, Chichester, Brisbane and Toronto, 1989.

9. Srivastava H. M., Owa S. (eds.). Current Topics in Analytic Function Theory. World Scientific Publishing Company, Singapore, New Jersey, London, and Hong Kong, 1992. https://doi.org/10.1142/1628

10. Uralegaddi B.A., Ganigi M.D., Sarangi S.M. Close to convex functions with positive coefficients. Stud. Univ. Babes-Bolyai Math., 1995, pp. 25-31.

H. M. Zayed, Department of Mathematics, Faculty of Science, Menofia University, Shebin Elkom 32511, Egypt (e-mail: hanaa_zayed42@yahoo. com)

M. K. Aouf, Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt (e-mail: mkaoufl27@yahoo.com)

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: maslinaOukm.edu.my)

Received 11.11.17

Н. М. гауеа, М. К. АоиГ, МаэНпа Багив

Некоторые подклассы аналитических функций с переменными аргументами в коэффициентах

Аннотация. В настоящей работе вводится и изучается класс V7Zg:V(n,X,a) аналитических функций с переменными аргументами в коэффициентах. Получены неравенства на коэффициенты, теоремы искажения с использованием дробного исчисления, радиусы почти выпуклости, звездности и выпуклости и извлечение квадратного корня для функций из класса V1Zg>rl(n, А, а). Рассмотрен интеграл свертки для функций в этом классе.

Ключевые слова: аналитические функции; произведение Адамара; операторы дробного исчисления; переменные аргументы коэффициентов; извлечение квадратного корня; интегральная свертка

Н. М. Zayed, Menofia University, Shebin Elkom 32511, Egypt (e-mail: hanaa_zayed42@yahoo.com)

M. K. Aouf, Mansoura University ( Mansoura 35516, Egypt (e-mail: mkaoufl27@yahoo.com)

Maslina Darus, Bangi 43600 Selangor D. Ehsan, Malaysia, tel.: 0060133882683 (e-mail: maslina@ukm.edu.my)

Поступила в редакцию 11.11.17