HANKEL DETERMINANT OF CERTAIN ORDERS FOR SOME SUBCLASSES OF HOLOMORPHIC FUNCTIONS Текст научной статьи по специальности «Математика»

URAL MATHEMATICAL JOURNAL, Vol. 8, No. 1, 2022, pp. 128-135

DOI: 10.15826/umj.2022.1.011

HANKEL DETERMINANT OF CERTAIN ORDERS FOR SOME SUBCLASSES OF HOLOMORPHIC FUNCTIONS

D. Vamshee Krishna

Department of Mathematics, Gitam School of Science, GITAM University, Visakhapatnam - 530 045, A.P., India [email protected]

D. Shalini

Department of Mathematics, Dr. B. R. Ambedkar University, Srikakulam - 532 410, A.P., India [email protected]

Abstract: In this paper, we are introducing certain subfamilies of holomorphic functions and making an attempt to obtain an upper bound (UB) to the second and third order Hankel determinants by applying certain lemmas, Toeplitz determinants, for the normalized analytic functions belong to these classes, defined on the open unit disc in the complex plane. For one of the inequality, we have obtained sharp bound.

Keywords: Holomorphic function, Upper bound, Hankel determinant, Positive real function.

1. Introduction

Let A represent a family of mappings f of the type

<x

f (z) = z + atZ t=2

in the open unit disc

U = {z € C : 1 > |z|},

and S is the subfamily of A, possessing univalent (schlicht) mappings. Pommerenke [17] characterized the rth-Hankel determinant of order n, for f with r, n € N, namely

HrM) =

an an+1 an+1 an+2

an+r-1 an+r

an+r-1 an+r

an+2r-2

(fl! = 1).

(1.1)

The Fekete-Szego functional [7] is obtained for r = 2 and n = 1 in (1.1), denoted by H2,i(f). Further, sharp bounds to the functional |H2,2(f)|, obtained for r = 2 and n = 2 in (1.1), are called as Hankel determinant of order two, given by

H2,2(f) =

a2 a3

0>3 0>4

— 02^4 — Ö"2-

In recent years, the estimation of an upper bound (UB) to |H2,2(/)| was studied by many authors. The exact estimates of |H2,2(/)| for the functions namely, bounded turning, starlike and convex functions, each one is a subfamily of S, symbolized as R, S* and K respectively and fulfilling the conditions

Re /'(z) > 0, Re

z/'(z)

> 0, Re <M +

z/''(z)

>0

/(z) J ' I /'(z)

in the unit disc U, were proved by Janteng et al. [9, 10] and the derived bounds are 4/9, 1 and 1/8 respectively. Choosing r = 2 and n = p + 1 in (1.1), we obtain Hankel determinant of second order for the p-valent function (see [20]), given by

H2,(p+1)(/)

ap+l ap+2 ap+2 ap+3

ap+1ap+3 — ap+2

The case r = 3 seems to be much tough than r = 2. Few papers were devoted for the study of third order Hankel determinant denoted as H3,1(/), with r = 3 and n = 1 in (1.1), namely

01 = 1 02 03

H3,1 (/) 02 03 04

03 04 05

Calculating the determinant, we have

H3,1(/) = 01(0305 - -04) + 02(0304 — 0205)

^3(0204 - a3). (1.2)

The concept of estimation of an upper bound for H3,1(/) was firstly introduced and studied by Babalola [3], who tried to estimate this functional in the classes R, S* and K, his results are as follows

(i) / € S* ^|H3,i(/)|< 16;

(ii) / € K ^ |H3,i(/)| < 0.714;

(iii) / €R^|H3,i(/)| < 0.742.

As a result of the paper by Babalola [3], mach research associated with the Hankel determinant of order 3 and 4, for specific subfamilies of holomorphic functions have been done (see [1-5, 11, 12, 15, 18, 19]). Motivated by the results obtained by the indicated authors, here we make an attempt to derive an upper bound to |H2,3(/)| = a3a5 — a2, |H3,1(/)|, when / belongs to the following new subfamilies of holomorphic functions.

Definition 1. A function /(z) € A is said to be in the class Rb(a), where b = 0 is a real number with a (0 < a < 1), if it satisfies the condition

( 2 2 , \

Re — - + -f (z)J > a, zeU.

It is observed that for b = 2 and for the values b = 2, a = 0, we have R(a), the class consisting of functions whose derivative has positive real part of order a (0 < a < 1) and R respectively.

Definition 2. A function /(z) € A is said to be in the class S*(a), where b is a non-zero real number with a (0 < a < 1), if it satisfies the condition

22 Re|1"6 + 6

zf'(z) №

> a, z € U.

For the values b = 2 and b = 2, a = 0, S* (a) reduces to S*(a), class consisting of starlike functions of order a (0 < a < 1) and S* respectively.

Definition 3. A function f (z) € A is said to be in the class Kb(a), where b = 0 is a real number with a (0 < a < 1), if it satisfies the condition

In particular for b = 2 and for the values b = 2, a = 0, Kb(a) reduces to K(a), the class consisting of convex functions of order a (0 < a < 1) and K respectively.

In proving our results, the following sharp estimates are needed, which are in the form of Lemmas hold good for functions possessing positive real part. Define the collection P of all functions g, each one called as Caratheodory function [6] of the form

<x

t

g(z) = 1 + X) ctzt,

t=i

which is holomorphic in U and Reg(z) > 0 for z € U.

Lemma 1 [8]. If g € P, then the estimate |c — ^Cjci-j| < 2 holds for i, j € N, with i > j and » € [0,1].

Lemma 2 [14]. If g € P, then the estimate |c — Cjci-j| < 2 holds for i, j € N, with i > j.

Lemma 3 [16]. If g € P, then |ct| < 2, for t € N, equality occurs for the function

h(z) = ] + *, z GU. 1—z

Lemma 4 [21]. If g € P, then |c2c4 — c31 < 4 — 1/2 ■ |c2|2 + 1/4 ■ |c2|3. In order to procure our results, we adopt the procedure framed through Libera and Zlotkiewicz [13].

2. Main results

Theorem 1. If

<x

f (z) = z + ^ a„zn € Rb(a),

n=2

where b is any real number with 0 <b < 1/(1 — a), for 0 < a < 1 then

4162(1 — a)2 №,i(/)| < -240-*

Proof. For

f (z) = z + ^ a„zn € Rb(a),

n=2

by virtue of Definition 1, we have

6(1 a\+{iUa)Z) 11 = <* 6(1 " «) + 2 {/'(*) " 1} = 6(1 " «)</(*)• (2.1)

Using the series representations for /'(z) and g(z) in (2.1), after simplifying, we get

o-n = 1, where t = 6(1 — a), n> 2. 2n

Putting the values of , for i € {2,3,4, 5} from (2.2), in H3,1(/), given in (1.2), we have

Hs,i(/)= t2

C2C4 tc2 c2 tc2c4 tc1c2c3 "60" ~ 216 ~ 64 ~ 160 + 96

(2.2)

(2.3)

On grouping the terms in the expression (2.3), we obtain

Hs,i(/)= t2

fe4(c2 - cj) _ C3 / _ tcic2\ tC'2 (c4 - c2)

160 64 V3 2 / 216

c2 / tgc3\ (189 -94t)c2c4 192 V4 2 J 8640

(2.4)

Applying the triangle inequality in (2.4), we get

Hs,i(/)

t2

+

t|C4||(C2 - c2)| , |C3|

160

+

64

C3

tci c2

2

+

t|C2||C4 - C2|

216

I C '21 192

c4 -

tCiC3

+

(189 — 94t)|c2||c4| 8640

(2.5)

Upon using the Lemmas 1-3 in the inequality (2.5), we obtain

41t2 41b2 (1 — a)2

|H3,i(f)l <

240

240

Remark 1. Choosing b = 2 and a = 0 in the inequality (2.6), it coincides with obtained by Zaprawa [22].

Theorem 2. If

(2.6) □

the result

/(z) = z + 0nzn € Rb(a),

n=2

where b is any real number with 0 < b < 1/(1 — a), for 0 < a < 1 then |H2,3(/)| < b2(1 — a)2/15.

Proof. Substituting the values of 03, 04, and 05 from (2.2) in H2,3(/), we have

H2,3(f ) = «3^5 - = t2

C2C4 c3

60 64

= t2

C2C4 C2C4 C2C4 c3

- - - _U - - -

60 64 64 64

= t2

C2C4 ~ C§ C2C4

64

960

(2.7)

, where t = b(1 — a).

Applying the triangle inequality in (2.7) and then using the Lemmas 3 and 4, after simplifying, we

get

.2,/ b2(1 — a)2

|H2,3(f)| = la3fl5 - «21 <

(2.8) □

2

OO

Remark 2. Choosing b = 2 and a = 0 in the inequality (2.8), it coincides with the result obtained by Zaprawa [21]. At this stage, the inequality in (2.8) becomes sharp for the function

g{Z) = —2-

Theorem 3. If

f (z) = z + ^ a„zn € (a),

n=2

where b is any real number with 0 < b < 1/(1 — a), for 0 < a < 1 then

|Hs,i(f)| <

6(1 -a) 12

[34 + b(1 - a)].

P r o o f. For

from the Definition 2, we have

f (z) = z + J2 anzn € S6*(a),

n=2

{6(1 — a) — 2} /(z) + 2zf'(z) 6(1 - a)f(z)

= g(z) ^ {b(1 - a) - 2} f (z) + 2zf'(z) = b(1 - a)f (z)g(z) (2.9)

Replacing f (z), f'(z) and g(z) with their equivalent series expressions in (2.9) and applying the same procedure as we carried in Theorem 1, we obtain

a2 = t-y, a3 = | (2c2 + tcf) , a4 = ^ (8c3 + 6icic2 + i2cf) ,

a5 = (48c4 + 32teic3 + 121(% + 12i2c?c2 + t3cf), where i = 6(1 - a). 384

(2.10)

Substituting the values of 0>2, «3, «4, and a5 from (2.10) in the functional given in (1.2), we get

- t4cf + 6t3c4c2 + 32t2cics - 36t2c2c2 - 144tc2c4

+192tciC2C3 - 72tc2 + 288C2C4 - 256c3

(2.11)

On grouping the terms in (2.11), we have

t \ 2

tc2

tc2

leofc.-^j^-fj+si^-^) +

tc2 \3

128(C2-f)(C4-^)- 256(C3-

8tcic2\2 16 )

(2.12)

On applying the triangle inequality in (2.12), we obtain

, , / t \ 2

I*U</>I £ (94)

128

c2

tc2i

160

c4

c2

tc2i

tc2

c4

+ 8t

c2

tc2i , 3

+

tci c3

+ 256

c3

8icic2 16

2

2

2

2

2

2

2

Further, the above inequality simplifies to

|H3,1(/)| <

[34 +t] =

6(1 - a) 12

[34 + b(1 — a)].

(2.13) □

Remark 3. Choosing b = 2 and a = 0 in the inequality (2.13), we see that it coincides with that of Zaprawa [22].

Theorem 4. If

/(z) = z + ^ 0nzn € Kb(a),

= z + >

n=2

where b is any real number with 0 < b < 1/(1 — a), 0 < a < 1 then

2

|H3,1(/)| <

b(1 — a)

[33 + 8b(1 — a)].

P r o o f. For

12A/15

<x

/(z) = z + 0nzn € Kb(a),

n=2

from Definition 3, we have

{b(l-a)-2}f(z)+2zf(z) b(l-a)f(z)

= g(z) ^ {b(1 — a) — 2} /(z) + 2z/'(z) = b(1 — a)/(z)g(z).

Applying the same procedure as we did in Theorem 1, we obtain

t

02 = t

05

1920

^p a3 = ^ (2c2 + tci) , «4 = ^ (8c3 + 6tcic2 + i2c?) , (48c4 + 32tc1 C3 + 12tc2 + 12t2c2C2 + t3c4) , where t = b(1 — a)

Further, we have

H3,1(/) =

552960

— t4c6 + 12t3 c1c2 + 48t2 c1c3 — 84t2c2c2 — 288tc2 C4

+288tC1 C2C3 — 32tc3 + 1152C2 C4 — 960c3

On grouping the suitable terms in the above expression, we have

H3,1(/) =

t2

552960

64i(c2 - M)3 + 384C4(C2 -t-f)+ 576C2(C4 - *-f)

+192 (C2 - f) (C4 _ _ 96003(03 - ^p) + 192ic2 (c2 - M)"

Applying the triangle inequality and then the Lemmas 1-3 in (2.14), we get

tc?'

tc2-

(2.14)

|H3,1(/)| <

t

[33 + 8t] =

6(1 - a) _ 12\/l5 .

[33 + 8b(1 — a)].

(2.15)

2

2

2

t

2

2

Remark 4. Choosing b = 2 and a = 0 in the inequality (2.15), we see that it coincides with the result obtained by Zaprawa [22].

3. Conclusion

The upper bounds to the fourth order Hankel determinants for all the above defined subclasses of analytic functions were derived.

Acknowledgements

The authors are highly grateful to the esteemed Referee(s) for a comprehensive reading of the manuscript and making valuable suggestions, leading to a better paper presentation.

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