On Janowski Type Harmonic Functions Associated with the Wright Hypergeometric Functions Текст научной статьи по специальности «Математика»

Vladikavkaz Mathematical Journal 2023, Volume 25, Issue 4, P. 91-102

YAK 517.53

DOI 10.46698/b2503-7977-9793-e

ON JANOWSKI TYPE HARMONIC FUNCTIONS ASSOCIATED WITH THE WRIGHT HYPERGEOMETRIC FUNCTIONS

G. Murugusundaramoorthy1 and S. Porwal2

1 School of Advanced Sciences, Vellore Institute of Technology, Vellore 632014, Tamil Nadu, India; 2 Ram Sahai Government Degree College, Bairi-Shivrajpur, Kanpur 209205, Uttar Pradesh, India E-mail: gmsmoorthy@yahoo.com, saurabhjcb@rediffmail.com

Abstract. In our present study we consider Janowski type harmonic functions class introduced and studied by Dziok, whose members are given by h(z) = z + 55=2 hnzn and g(z) = ^1 gnzn, such that = {/ = h + g € H : ^fffi1 -< i^f; (-G<F<G<1, with gi = 0)}, where ®Hf(z) =

zh'(z) — zg'(z) and z £ U = {z : z £ C and |z| < 1}. We investigate an association between these subclasses of harmonic univalent functions by applying certain convolution operator concerning Wright's generalized hypergeometric functions and several special cases are given as a corollary. Moreover we pointed out certain connections between Janowski-type harmonic functions class involving the generalized Mittag-Leffler functions. Relevant connections of the results presented herewith various well-known results are briefly indicated.

Keywords: harmonic functions, univalent functions, Wright's generalized hypergeometric functions. AMS Subject Classification: 30C45, 30C55.

For citation: Murugusundaramoorthy, G. and Porwal, S. On Janowski Type Harmonic Functions Associated with the Wright Hypergeometric Functions, Vladikavkaz Math. J., 2023, vol. 25, no. 4, pp. 91102. DOI: 10.46698/b2503-7977-9793-e.

1. Introduction

A continuous complex-valued function f = u + iv defined in a simply-connected domain D is said to be harmonic in D if both u and v are real harmonic in D. In any simply-connected domain D, we can write f = h + g, where h and g are analytic in D and are commonly denoted by H. In 1984 Clunie and Sheil-Small [1] introduced a class of complex — valued harmonic maps f which are univalent and sense — preserving in the open unit disk U = {z : z € C and |z| < 1}. The function f e can be represented by f = h+g, is given by

tx> tx>

j{z) = h{z) +J(z) = z + J2hnZn + J2 9nZn,

where

n=2 n=1

h(z)= z + £hnzn, g(z) = J29nzn, \gi\ < 1 (1.1)

z + > hn z

n=2 n=1

© 2023 Murugusundaramoorthy, G. and Porwal, S.

are analytic in the open unit disk in U. They also proved that the function f = h + g € SH is locally univalent and sense preserving in U, if and only if |h'(z)| > |g'(z)|, V z € U. For more basic study one may refer Duren [2] and Ahuja [3]. It is worthy to note that if g(z) = 0 in (1.1), then the class SH reduces to the familiar class S of analytic functions. For this class f (z) may be expressed as of the form

ro

f (z) = z + £ h„zn. (1.2)

n=2

Further, we suppose that SH subclass of SH consisting of function f € SH of the form (1.1) with g1 = 0 and is given by

/(z) = h{z) + g{z) =z + Y, hnzn + J] 9nZn,

n=2 n=2

where

ro ro

h(z) = z + h„zn, g(z) = £ grazra, gi = 0.

n=2 n=2

Now, we let KH, STH and CH denote the subclasses of S0 of harmonic functions which are respectively convex, starlike and close-to-convex in U. Also let To be the class of sense preserving, typically real harmonic functions f = h + g in SH. For detailed study of these classes one may refer to [1, 2].

Now, we recall the subclass of S^n consisting of functions f = h + g, so that h and g are of the form

ro ro

h(z)= z -J]|h„|zn and g(z) = £ |gn|zn, (1.3)

n=2 n=1

which has been introduced and studied extensively by Silverman [4].

Let a € C, ((a/A) = 0,-1, -2,... ; i = 1, 2,...,p) and ((6*/Bj) = 0, -1, -2,...; i = 1, 2,..., q), for Aj > 0 (i = 1,... ,p), Bj > 0 (i = 1,..., q) with 1 + ^?=1 Bj - EPU Aj ^ 0 the Wright's generalized hypergeometric functions [5] is defined by

P^q

(a^ Aj)1,p ; z

(bj,Bj )1,q

P

ro n r(aj + nAj)zn

E^-> (i-4)

n=o n r(6j + nB^n!

j=1

which is analytic for suitable bounded values of |z| (see also [6, 7]). The generalized Mittag-Leffler, Bessel-Maitland and generalized hypergeometric functions are some of the important special cases of Wright's generalized hypergeometric functions and for their details one may refer to [7-9].

For Aj > 0 (i = 1,... ,p), Bj > 0, 6j > 0 (i = 1,... , q) with 1 + £?=1 Bj - E?^ A ^ 0 and Cj > 0 (i = 1,..., r), Dj > 0, dj > 0 (i = 1,..., s) with 1 + £S=1 Dj - £r=1 Cj ^ 0, we define Wright's generalized hypergeometric functions

P^q

Aj)1,p ; z

(bj,Bj )1,q

P

ro n r(aj + nAj)zn

£

j=1

q

ra=o EI r(6j + nBj)n! j=1

ro

ro

and

r

ro n r(cj + nCj)z'

if: f , • I 1 .. «-^

(cj ,Cj) 1,r (dj,Dj)1,s '

E¥--(L5)

n=o r(dj + nDj)n!

j=1

with

r

EE r(|cj| + nCj)/r|cj|

¥-< 1-

n r(dj + nDj)/r(dj) i=1

We consider a harmonic univalent function

F(z) = H(z) + G(z) € Sh, (1.6)

where

q

n r(6j) r . .. n ro

(I'M, Ai)itP _

(bj,Bj )1,q

= zl-±-

n r(aj) i=1

z + ^ zn (1.7)

n=2

and

s

n r(dj)

aJ2<nZn, H<1, (1.8)

<8(z) = az -

n r(cj) i=1

(ci, Ci) 1,r

(dj,Dj)1,s '

ro

n=1

and and are given by

P

n r(a + (n - 1)Aj)/r(aj)

= --(1.9)

n (r(bj + (n - 1)Bj)/r(6j))(n - 1)! i=1

and

r

= -• (1.10)

n r(cj + (n - 1)Cj)/r(cj) ¿=i_

II (r(dj + (n - 1)Dj)/r(dj))(n - 1)! i=1

From (1.9) and (1.10), we have for n € N = {1, 2,... }

P

11 r(|aj| + (n - 1)Aj)/r(|aj|)

\6n\ < -= Vn (Ln)

n (r(bj + (n - 1)Bj)/r(6j))(n - 1)! i=1

and

r

11 r(|cj| + (n - 1)Cj)/r(cj)

ICnl < -j—^- = rin. (1.12)

n (r(dj + (n - 1)Dj)/r(dj))(n - 1)! i=1

For some fixed value of j € No = N U {0} and for

L[Bf II Df ^ Cf

¿=1

¿=1

¿=1

¿=1

we denote

P^q

(|ai| + jAi, Aj)1,p (|bi | + jBi,Bi) 1,q

; 1

; r

(|ci| + jCi,Ci)1,r (|di| + jDi ,Di)1s

r

provided that

q p — 1 s r — i

" £ N + ^ > 9 > o +3-

¿=1 ¿=1

¿=1 ¿=1

2 2

Making use of (1.11), (1.12) and (1.13), we have

(1.13)

(1.14)

Y1(n - j)

n=1+j

n r(b)

_ i= 1_

J "n p

n r(W)

¿=1

PWq

and

~ n r(d)

(n - fa vn = —

n=1+j n r(|Cj|)

r ws

¿=1

provided that (1.14) holds true.

The convolution of two functions f (z) of the form (1.1) and F(z) of the form

tx> tx>

n I x „ ~n

F(z) = z + hnzn + J2 gnz

n=2 n=1

be given by

(f * F)(z) = f (z) * F(z) = z + ^ hnhnzn + ^ gngnz

n=2 n=1

Now, we introduce a convolution operator Q(p, q,r, s) as

Q(p, q, r, s)f (z) = f (z) * F(z) = h(z) * H(z) + g(z) * G(z),

(1.15)

(1.16)

where / = h + g and $(z) = Sj(z) + <S(z) given by (1.1) and (1.6) respectively. Hence

s)f (z) = z + ^ 0nhnzn + ^ ^Cngnz

n n n

n=2 n=1

(1.17)

(1.18)

(1.19)

(1.20)

We consider the following classes of functions due to Dziok [10]. Let STh(F, G) denote the class of functions f € whose members are given by (1.1), such that

yFH{F,G) = |f = h + g € H : ^jffi ("G<F<G<1, with g1=0)|, (1.21)

q

P

s

r

1

DO

CO

where S)fff(z) = zh'(z) — zg'(z) and z € U, where H denote the class of harmonic functions in the unit disc U.

Moreover, let us define

CV h (F, G) := {/ € ST h : DH / € ST H (F, G)}.

We should notice that the class

ST (F, G) := ST H (F, G) n A

was introduced by Janowski [11]. The classes

ST h (a) := ST h (2a - 1,1) and CV h (a) := CV h (2a - 1,1)

were investigated by Jahangiri [12, 13] also see [4]. Finally, the classes

ST h := ST h (0) and CV h := CV h (0)

are the classes of functions / € SH which are starlike in U and STH(F, G) C STH,

CV H (F, G) C CV H .

Lately, Dziok [10] gave the following necessary and sufficient coefficient condition for

/ € ST H (F, G).

Lemma 1 [10]. Let / € H be assumed as in (1.1), then / € STH(F, G) if

œ

" 'in

n=2

Ë ([n(1 + G) - (1 + F)] |h„| + [n(1 + G) + (1 + F)] |gra|) < G - F, (1.22)

where h1 = 1, g1 =0 and (-G < F < G < 1).

Lemma 2 [10]. Let f £ H be assumed as in (1.3) and f € ST^(F, G) if and only if

œ

" 'in

n=2

Ë ([n(1 + G) - (1 + F)] |h„| + [n(1 + G) + (1 + F)] |gra|) < G - F, (1.23)

where h1 = 1, g1 =0 and (-G < F < G < 1).

Remark 1. In [10], it is also shown that f = h + g be given by (1.3) is in the family ST^(F,G), if and only if

roro

J>(1 + G) - (1 + F)] |h„| + J>(1 + G) + (1 + F)] |gra| < G - F, (1.24)

n=2 n=1

where h1 = 1, |g11 < 1 and (-G < F < G < 1).

Moreover we note that, if f € STH(F, G), then

G-F G-F

^Kn(l + G)-(1 + F)' and ^^n(l + G) + (l + Fyn>1-

The application of the special functions on Geometric Function Theory always attracts researchers various kinds of special functions for example hypergeometric functions [14-16], confluent hypergeometric functions [17], generalized hypergeometric functions [5, 18]. Wright functions [19-22], Fox-Wright functions [5, 23], Mittag-Leffler functions [24] generalized Bessel functions [25], have rich applications in analytic and harmonic univalent functions. Motivated with the work of [21, 26, 27], we obtain some inclusion relation between the classes STh(F, G), K0, and STH, or CH by applying he convolution operator Q.

2. Mapping Properties of H Related with Convolution Operator Q

In order to establish our main results we shall require the following lemmas. Lemma 3 [1]. If f = h + g € where h and g are given by (1.1) with g1 = 0, then

171 n +1 . . n — 1

\hn\ < —2—, |gn| < —•

Lemma 4 [1]. Let f = h + g € STH or , where h and g are given by (1.1) with

g1 = 0. Then

IM<(2" + 1>(" + 1) a,d W^-1«"-1»

6

6

Theorem 1. Let - ELi N + V > I and £i=i ^ - E[=i M + V > I'

the inequality

n r(bi)

n r(|aj |)

¿=1

{(1 + G)p*2 + (3 + 4G — F) + 2 (G — F) (p*0 — 1)}

(2.1)

n r(dj)

+ M

¿=1

n r(|ci|)

¿=1

{(1 + G) r^2 + (3 + 2G + F) r< 2 (G — F)

holds. Then C STh(F,G).

< Let f = h + g € K0, where h and g are given by (1.1) with g1 = 0. We have to prove that Q(f) € STh(F, G), where Q(f) is defined by (1.20). To prove Q(f) € STh(F,G), in view of Lemma 1, it is sufficient to prove that P1 < G — F, where

P1 = £[n(1 + G) — (1 + F)] |0nhn| + J>(1 + G) + (1 + F)] |Zngn|. (2.2)

n=2 n=2

By using Lemma 3

x x

+ 1) [n(1 + G) — (1 + F)] |<y + — 1) [n(1 + G) + (1 + F)] !Cn|

2

n=2 x

n=2

^ {(n — 1)(n — 2)(1 + G) + (n — 1) [3 + 4G — F] + 2 (G — F)} v,

n=2

+

H 2

{(1 + G)(n — 2) + (3 + 2G + F)} nn

n=2

q

n r(bi)

-{(l + G)p^ + [3 + 4G-F]p^ + 2(G-F)(p^-l)}

n r(|ai|)

¿=1

s

n m)

»—1 f/1 I I /9 I or< I Z7\ iTrll

+ M

{(1 + G) r ^2 + (3 + 2G + F) r

nr(|c, |)

¿=1

< G — F

by the given hypothesis. This completes the proof of the Theorem 1.

x

x

x

The result is sharp for the function

ro / 11 \ ro /

n=2

n=2

2

zn >

Theorem 2. Let Eh h ~ ELi N + ^ > I and ^ " ELi M + V > 3- ^

the inequality q

n r(bi)

-{2 (1 + G) + (15G + 2F + 13) + (24G -9F + 15)

n r(|a |) i=1

+ 6(G - F)(- 1)} + M

n r(di)

(2.3)

i=1

n r( | Ci |) i=1

{2 (1 + G) r+ (9G + 2F + 11) rtf

2

r ^s

+ 3(2G + F + 3)r< 6 (G - F)

holds, then Q(STH) C (F, G) and Q(C°) C STh(F,G).

< Let f = h + g € STH(or, ), where h and g are given by (1.1) with g1 = 0, we need to prove that Q(f) € STH(F, G), where Q(f) is defined by (1.20). In view of Lemma 1, it is sufficient to prove that P1 ^ 1 — y, where P1 is given by (2.2). Now using Lemma 4, we have

1

6

_ n=2

x [n(1 + G) + (1 + F)] |C„|

+ 1)(2n + 1)[n(1 + G) - (1 + F)] |0ra| + H ^(n - 1)(2n - 1)

n=2

ro

Y^ {2(1 + G)(n - 1)(n - 2)(n - 3)

n=2

+ (15G + 2F + 13)(n - 1)(n - 2) + (24G - 9F + 15)(n - 1) + 6(G - F)} vra

ro

^ {2(1 + G)(n - 1)(n - 2)(n - 3) + (9G + 2F + 11)(n - 1)(n - 2)

+

M

n=2

+ (6G + 3F + 9)(n - 1)} nn

n r(bi)

-{2(1 + G) + (15G + 2 F + 13) p^2q

n r(|0i|)

.¿=1

+ (24G - 9F + 15) p^ + 6 (G - F) (p*0 - 1)}

+

M

n r(di)

^-{2(1 + G)r^

n r(|ci|)

.¿=1

+ (9G + 2F + 11) r ^2 + 3 (2G + F + 3) rtfj}

< G - F

s

ro

ro

6

by the given hypothesis. Thus, the proof of Theorem 2 is established.

The result is sharp for the function

_ i I 1 ?2 _L 1 ?3

f(„\ _ ^ 2 '6 I 2 6 ^

(1-z)3 + (1-z)3 * ^

In our next theorem, we establish connections between STH(F, G). Theorem the inequality

Theorem 3. Let £?=i bt - H + > i and di _ |Ct| + r^ > i. If

n r(bi) n r(dj)

- - 1) + M ^-< 1 (2.4)

nr(|at|) n r(|cl|)

¿=1 ¿=1

holds, then Q(STh(F,G)) C STh(F,G).

< Let f = h + g € STH(F, G) be given by (1.1) with |g1| < 1. We have to prove that P2 < G — F, where

x x

P2 = J>(1 + G) — (1 + F)] |0n hn| + M ^[n(1 + G) + (1 + F)] |Zngn|. (2.5)

n=2 n=1

Now, using Remark 1, we have

P2 < (G - F) + (G - F) ^nn

= (G - F)

X X

Vn + (G - F) a

n=2 n=1

/ q s x

' n r(bi) n r(di) x

^-W -1) + M ^-

\ n r(|0i|) n r(|Ci|) ,

\i=1 i=1 /

< G - F

by the given hypothesis, this completes the proof of Theorem 3. The result is sharp for the function

=2" J (»(i+g)-(i+F)) w+1 (n(i+G) + a + f))

where

x x

J]|Xn| +J2 |yn| = 1- >

n=2 n=1

s

3. Some Consequences Related with Mittag-Leffler Functions

If we let p = q = r = s = 1 and a1 = A1 = c1 = C1 = 1 in (1.6), then W(z) reduces to a harmonic univalent function E(z) involving the following generalized Mittag-Leffler functions

as _

(3.1)

E(z) = zr(bi) El;1Bl [z] + azr(di) eJ;1Di [z],

where

e111bI [z] = 1*1

(1,1) (b1,B1)

E

n=0

r(b1 + nB1)

n

z

z

and

" (1,1)

EdlU [z] = 1Ф1

(di,Di) ; z

0 r(di + nDi)

n=0

With these specializations, the convolution operator Q(p, q,r, s) reduces to the operator b1; B1; d1; D1), which is defined as

$(bi; Bi; di; Di)f (z) = f (z) * E(z) = h(z) * z^b^E^[z] + ag(z) * z^d^E^[z]. (3.2)

For these specific values of p = q = r = s = 1 and ai = Ai = ci = Ci = 1, Theorems 1-3 yield the following results.

Corollary 1. If the inequality

3,i , in , АП tt.2,1 , n(n r\ ^i,i

?1 -1 ?

(3.3)

r(6i) {(1. + G) E3;+2Bi(1) + (3 + 4G - F) E2;+Bi;Bi(1) + 2 (G - F)(Ei,iBi - ^ } + M ВД) {(1 + G) Edii+2Di)Di (1) + (3 + 2G + F) eJ+DiD (1)} < 2 (G - F)

holds. Then ) C STh(F,G).

Now, we state new inclusion results for Janowski-type harmonic functions due to Dziok [10] without proof.

Corollary 2. If the inequality

r(6i) {2 (1 + G) <+3^! (1) + (15G +2F +13) e3;+2Bi)B1(1)

+ (24G - 9F + 15) E^+Bi Bi (1) + 6 (G - F) (E^ - 1)}

+ и r(d i ) {2 (1 + G) Ed;+3Di ,Di (1) + (9G +2F +11) Ed;1+2Di,Di (1) + 3 (2G + F + 3) Edi+DiD (1)} < 6 (G - F)

(3.4)

holds. Then $(STH) С STh(F,G), and Ф(СН) С STh(F,G). Corollary 3. If the inequality

r(b i) { (eJ;,Bi - 1)} + И r(d i) (EJ;^DJ < 1 (3.5)

holds. Then $(ST^(F,G)) С STH(F,G).

Remark 2. If we put p = q = r = s = 1, ai = ci = 1, Ai = Ci = 0 and а = 1, then

жы-z + v_^_zra + v r(dl)

n=2 r(b1 + B1(n - 1))(n - 1)! r(d1 + A(n - 1))(n - 1)!

and results of Theorems 1-3 gives to new inclusion results for Janowski-type harmonic functions due to Dziok [10].

Remark 3. If we put p = r = 2, q = s = 1 and A1 = A2 = B1 = C1 = C2 = D1 = 1 and a = 1, then

yw=z+v {;hi\n~i la2)n~) z-+eM*-1 iC2)ra;l,

n=2 (ft1)n-1 (n - 1)! (d1)n-1 (n - 1)!

and results of Theorems 1-3 yields the new results for the subclasses of Janowski-type harmonic functions due to Dziok [10].

zn

n

Remark 4. By taking F = 2a — 1 and G = 1 one can get the inclusion results for the subclasses STH(2a — 1,1) = STH(a) and CVH(2a — 1,1) = CVH(a) defined and studied by Jahangiri [12, 13].

Concluding Remark. By defining

CVh(F, G) := f € STH : DHf € STH(F,G)} due to Dziok[10] as in Lemma 1 we state the following result: A function f € CVH(F, G) if

EM-G^F-+-G^F-l9nl) (3'6)

n=2

where h1 = 1; g1 =0 and (—G < F < G < 1). Proceeding as in above results we can obtain analogous inclusion results for the function class CVH (F, G) we left this as an exercise to interested readers.

Acknowledgement. We authors record our sincere thanks to the referees for their valuable comments to revise the paper in present form.

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Received February 2, 2021

Gangadharan Murugusundaramoorthy

School of Advanced Sciences, Vellore Institute of Technology,

Vellore 632014, Tamil Nadu, India,

Professor of Mathematics

E-mail: gmsmoorthy@yahoo. com

https://orcid.org/0000-0001-8285-6619

Saurabh Porwal

Ram Sahai Government Degree College, Bairi-Shivrajpur, Kanpur 209205, Uttar Pradesh, India, Assistant Professor of Mathematics E-mails: saurabhj cb@rediffmail. com https://orcid.org/0000-0003-0847-3550

Владикавказский математический журнал 2023, Том 25, Выпуск 4, С. 91-102

О ГАРМОНИЧЕСКИХ ФУНКЦИЙ ТИПА ЯНОВСКОГО, СВЯЗАННЫХ С ГИПЕРГЕОМЕТРИЧЕСКИМИ ФУНКЦИЯМИ РАЙТА

Муругусундарамурти Г.1, Поруал С.2

1 Школа передовых наук, Технологический институт Веллора, Индия, Тамил Наду, 632014, Веллор;

2 Государственный колледж Рам Сахай, Индия, Уттар-Прадеш, 209205, Канпур, Баири-Шивраджпур E-mail: gmsmoorthy@yahoo.com, saurabhjcb@rediffmail.com

Аннотация. В настоящей работе мы рассматриваем класс гармонических функций типа Яновского, введенный и изученный Дзиоком, члены которого задаются формулой h(z) = z + Ю=2 hnzn, g(z) =

Ею n »

n=1 gnz такой, что

= = + (-G<F<G<1, gi = 0)J ,

где DH f (z) = zh'(z) — zg'(z), z £ U = {z : z £ C и |z| < 1}. Мы изучаем связь между этими подклассами гармонических однолистных функций, применяя определенный оператор свертки, касающийся обобщенных гипергеометрических функций Райта, и в качестве следствия приводятся несколько частных случаев. Кроме того, мы указали на определенные связи между классом гармонических функций типа Яновского, включающими обобщенные функции Миттаг-Леффлера. Кратко указаны соответствующие связи представленных результатов с различными известными результатами.

Ключевые слова: гармонические функции, однолистные функции, обобщенные гипергеометрические функции Райта.

AMS Subject Classification: 30C45, 30C55.

Образец цитирования: Murugusundaramoorthy G. and Porwal S. On Janowski Type Harmonie Functions Associated with the Wright Hypergeometrie Functions // Владикавк. матем. журн.—2023.—Т. 25, № 4.—C. 91-102 (in English). DOI: 10.46698/b2503-7977-9793-e.