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Miskolc Mathematical Notes HUe-ISSN1787-2413
Vol. 20 (2019), No. 2, pp. 1245–1260 DOI:10.18514/MMN.2019.3046
SOMEPROPERTIESOFANALYTICFUNCTIONSASSOCIATED
WITHFRACTIONALq-CALCULUSOPERATORS
H. M. SRIVASTAVA,M.K.AOUF,ANDA.O.MOSTAFA
Received 24 September, 2019
Abstract. By applying a fractional q-calculus operator, we define the subclasses S˛.;ˇ;b;q/
n
andG˛.;ˇ;b;q/ofnormalizedanalyticfunctionswithcomplexorderandnegativecoefficients.
n
Among the results investigated for each of these function classes, we derive their associated
coefficient estimates, radii of close-to-convexity, starlikeness and convexity, extreme points, and
growth and distortion theorems.
2010 Mathematics Subject Classification: 26A33; 30C45; 33D05
Keywords: analyticfunctions, fractional q-calculus operators, q-gamma functions, starlike func-
tions of complex order, convex functions of complex order, close-to-convex functions of com-
plex order, coefficient estimates, radii of close-to-convexity, starlikeness and convexity, extreme
points, growth and distortion theorems
1. INTRODUCTION AND DEFINITIONS
Here, in this paper, we denote by A.n/ the class of functions of the following
normalized form:
1
X k
f.´/D´C ak´ .n2NI NWDf1;2;3;g/; (1.1)
kDnC1
which are analytic in the open unit disk U centered at the origin (´ D 0) in the com-
plex ´-plane. We write A.1/ D A. We also denote by T .n/ the subclass of A.n/
consisting of functions of the form:
1
X k
f.´/D´ a ´ .a =0I k =nC1I n2N/: (1.2)
k k
kDnC1
In our investigation, we make use of various operators of q-calculus and fractional
q-calculus. For this purpose, we refer the reader to the various definitions, notations
andconventions,whichareconsiderablydetailedinourearlierpaper(see,fordetails,
[22]; see also [8]).
c
2019MiskolcUniversityPress
1246 H. M. SRIVASTAVA,M.K.AOUF,ANDA.O.MOSTAFA
For a fixed 2C,asetDiscalleda-geometricsetifandonlyifboth´2Dand
´2D. For a function f defined on a q-geometric set, we make use of Jackson’s
q-derivative and q-integral .0 < q < 1/ of a function on a subset of C, which are
already introduced in several earlier investigations (see, for example, [2], [4], [6], [8],
[9], [10], [14], [15], [16], [17], [21], [22] and [25]).
Now, for a complex-valued function f.´/; we introduce the fractional q-calculus
operators as follows (see, for example, [12] and [13]; see also [1]).
Definition 1 (Fractional q-integral operator). The fractional q-integral operator
I oforderisdefined,forafunctionf.´/, by
q;´
I f.´/DD f.´/D 1 Z ´.´ tq/ f.t/d t .>0/; (1.3)
q;´ q;´ ./ 1 q
q 0
where the function f.´/ is analytic in a simply-connected region of the complex
´-plane containing the origin. Here, and elsewhere in this paper, the q-binomial
.´ tq/ 1 is given by " #
1 1 k
1 Y 1 .tq´ /q
.´ tq/ 1 D´ 1 Ck 1
kD0 1 .tq´ /q
1 1
D´ 1˚0.q I Iq;tq ´ /: (1.4)
Remark 1. The q-hypergeometric series 1˚0.I Iq;´/ is known to be single-
valued when arg.´/ < (see, for example, [8]). Therefore, the q-binomial .´
j j
tq/ 1 in (1.4) is single-valued when
ˇ ˇ
ˇ ˇ ˇ ˇ
tq
ˇ 1 ˇ ˇ ˇ
arg tq ´ <; <1and arg.´/ <:
ˇ ˇ ˇ ˇ j j
ˇ ´ ˇ
Definition 2 (Fractional q-derivative operator). The fractional q-derivative oper-
ator D of order .05<1/isdefined,forafunction f.´/, by
q;´
D f.´/DD I1 f.´/D 1 D Z ´.´ tq/ f.t/d t; (1.5)
q;´ q;´ q;´ .1 / q q
q 0
where f.´/ is suitably constrained and the multiplicity of .´ tq/ is removed as
in Definition 1.
Definition 3 (Extended fractional q-derivative operator). Under the hypotheses of
Definition 2, for a function f.´/; the fractional q-derivative of order is defined by
D f.´/DDm Im f.´/ .m 15<1Im2N/: (1.6)
q;´ q;´ q;´
Clearly, we have
.nC1/
n q n
D ´ D ´ .=0I n> 1/:
q;´ .nC1 /
q
SOMEPROPERTIES OF ANALYTIC FUNCTIONS... 1247
Now, by using the operator D ; we define (for 1 < < 2; 0 < q < 1 and
q;´
´ 2 U;) a q-differintegral operator ˝ W T .n/ ! T .n/ as follows (see [12] and
q;´
[13]):
1
.2 / X
q k
˝ f.´/D ´ D f.´/D´ A .;k/a ´ (1.7)
q;´ ./ q;´ q k
q kDnC1
where
.kC1/ .2 /
A .;k/D q q (1.8)
q .2/ .kC1 /
q q
and D f.´/ in (1.7) represents, respectively, the fractional q-integral of f.´/ of
q;´
order . 1<<0/andthefractionalq-derivativeoff.´/oforder .05<2/
(see, for details, [7,18–20]). We note that some interesting special and limit cases
of (1.7) were investigated in the earlier works by Owa and Srivastava [11] and by
Srivastava and Owa (see [23] and [24]).
Remark 2. From (1.3), (1.7) and (1.8), we find that
.2C/ .2C/
q q
˝ f.´/D ´ D f.´/D ´ I f.´/
q;´ .2/ q;´ .2/ q;´
q q
1
X k
D´ A . ;k/a ´ ; (1.9)
q k
kDnC1
where
.kC1/ .2C/
A . ;k/D q q .>0I 00I b2CDCnf0g/
n
if it satisfies the following condition:
ˇ 0 ˇ
1
ˇ ˇ
ˇ .1 ˛/´Dq ˝ f.´/ C˛´Dq ´Dq ˝ f.´/ ˇ
ˇ1 B q;´ q;´ Cˇ
ˇ B Cˇ
ˇ 1 ˇ<ˇ: (1.11)
b @ .1 ˛/˝ f.´/C˛´Dq ˝ f.´/ A
ˇ q;´ q;´ ˇ
ˇ ˇ
Some of the interesting particular cases of the function class S˛.;ˇ;b;q/ are
n
being recorded below:
(i) S˛.;1;b;q/DS˛.;b;q/ (see[12]);
n n
(ii) S˛.0;ˇ;b;q/DS˛.ˇ;b;q/,where
n n
S˛.ˇ;b;q/WD(f Wf 2T.n/ and
n
1248 H. M. SRIVASTAVA,M.K.AOUF,ANDA.O.MOSTAFA
ˇ ˇ
2 2 ! )
ˇ1 .1 ˛/´Dqf.´/C˛´ D f.´/ ˇ
ˇ q 1 ˇ<ˇ :
ˇ ˇ
˛ ˇb .1 ˛/f.´/C˛´Dqf.´/ ˇ
(iii) lim Sn.ˇ;b;q/DSn.b;˛;ˇ/ (see[3]);
q!1
(iv) S0.;ˇ;b;q/DS.;ˇ;b;q/,where
n n
ˇ ˇ
( ! )
ˇ1 ´D ˝ f.´/ ˇ
S .b;˛;ˇ/WD f Wf 2T .n/ and ˇ q q;´ 1 ˇ<ˇ I
n ˇ ˇ
ˇb ˝q;´f.´/ ˇ
(v) lim Sn.;ˇ;b;q/DKn.;b;ˇ/ (see[5]withp D1);
q!1
(vi) S1.;ˇ;b;q/DCn.;ˇ;b;q/,where
n
( ˇ 2 !ˇ )
ˇ1 ´D ˝ f.´/ ˇ
ˇ q q;´ ˇ
Cn.;ˇ;b;q/WD f Wf 2T .n/ and ˇ 1 ˇ<ˇ :
ˇb D ˝ f.´/ ˇ
q q;´
Definition 5. A function f.´/ 2 T .n/ is in the function class
˛
Gn.;ˇ;b;q/ .<2I 05˛51I 00/
if it satisfies the following condition:
ˇ ˇ
ˇ1 2 ˇ
ˇ Dq ˝ f.´/ C˛´D ˝ f.´/ 1 ˇ<ˇ: (1.12)
ˇb q;´ q q;´ ˇ
Wechoosetonotethefollowing special case of the function class G˛.;ˇ;b;q/:
n
(i) G˛.0;ˇ;b;q/DG˛.ˇ;b;q/,where
n n
ˇ ˇ
˛ ˇ1 2 ˇ
Gn.ˇ;b;q/D f Wf 2T .n/ and ˇ Dqf.´/C˛´Dqf.´/ 1 ˇ<ˇ I
˛ ˛ ˇb ˇ
(ii) G .;1;b;q/DR .;b;q/ (see[13]);
n n
(iii) G˛.0;ˇ;b;q/DRn.˛;ˇ;b;q/ (see[13]);
n
(iv) lim G˛.0;ˇ;b;q/DRn.˛;ˇ;b/ (see[3]).
q!1 n
For each of the above-defined general function classes S˛.;ˇ;b;q/ and
n
G˛.;ˇ;b;q/ of analytic functions with complex order and negative coefficients,
n
we propose here to investigate the associated coefficient estimates, radii of close-
to-convexity, starlikeness and convexity, extreme points, and growth and distortion
theorems.
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