SECOND HANKEL DETERMINANT FOR BI-UNIVALENT FUNCTIONS ASSOCIATED WITH Q-DIFFERENTIAL OPERATOR Текст научной статьи по специальности «Математика»

DOI: 10.17516/1997-1397-2022-15-5-663-671 УДК 517.9

Second Hankel Determinant for Bi-univalent Functions Associated with ^-differential Operator

Mallikarjun G. Shrigan*

Bhivarabai Sawant Institute of Technology and Research Pune, Maharashtra State, India

Received 21.09.2021, received in revised form 10.03.2022, accepted 20.07.2022 Abstract. The objective of this paper is to obtain an upper bound to the second Hankel determinant denoted by H2(2) for the class S*(a) of bi-univalent functions using q-differential operator. Keywords: Hankel determinant, bi-univalent functions, q-differential operator, Fekete-Szego functional.

Citation: M.G. Shrigan, Second Hankel Determinant for Bi-univalent Functions Associated with q-differential Operator, J. Sib. Fed. Univ. Math. Phys., 2022, 15(5), 663-671. DOI: 10.17516/1997-1397-2022-15-5-663-671.

Let A denote the class of functions f (z) of the form:

f (z) = z + J2 ak zk (ak > 0; n e N = {1, 2, 3, •••}), (1)

k=n

which are analytic and univalent in the open unit disk given by

U = {z : z e C and |z| < 1}.

The Koebe one-quarter theorem [5] ensures that the image of U under every univalent function f e S contains a disk of radius 4. Hence every function f e S has an inverse f-1, which is defined by

f-1(f (z)) = z (z e U)

and

f-1(f(w)) = w <ro(f);ro(f) > 4

where

f-1(w) = w — a^w2 + (2a\ — a^)w3 — (5a| — 5a2 a3 + a4)w4 + • • • .

A function f (z) e A is said to be bi-univalent in U if both f and f-1 are univalent in U. We denote by E the class of all functions f which are bi-univalent in U and are given by the Taylor-Maclaurin series expansion (1). The behavior of the coefficients is unpredictable when the biunivalency condition is imposed on the function f e A. A systematic study of the class E of bi-univalent function in U, which is introduced in 1967 by Lewin [12]. For a brief history and interesting examples of functions which are in (or which are not in) the class E, together with various other properties of the bi-univalent function class E, one can refer to the work of

* [email protected] © Siberian Federal University. All rights reserved

Srivastava et al. [21] and references therein. Ever since then, several authors investigated various subclasses of the class E of bi-univalent functions. For some more recent works see [22-27]. The class of bi-starlike functions is introduce by Brannan and Taha [2] (see also [14]). For 0 < a < 1, a function f £ A is in the class S*(a) of bi-starlike function of order a if both f and f-1 are starlike in U and obtained estimates on the initial coefficients conjectured that \a2\ < %/2. It may be noted that for a = 0, q —> 1-, S*(a) = S*, the familiar subclass of starlike functions in U.

For the univalent function in the class A, it is well known that the nth coefficient an is bounded by n. The bounds for the coefficients gives information about the geometric properties of these functions. For example growth and distortion properties of normalized univalent function are obtained by using the bounds of its second coefficient a2. In 1966, Pommerenke [15] define the Hankel determinant of f for q ^ 1 and n ^ 1 as

Hq (n)

an an+i

an+1 an+2

n+q-1

an+q

an+q+1 an+q

a

n+2q-2

(2)

A good amount of literature is available about the importance of Hankel determinant. It plays an important role in the study of singularities as well as in the study of power series with integral coefficients ([3,4]). In 1916, Bieberbach proved that if f € S, then |a2 — a3\ < 1. In 1933, Fekete and Szego [5] proved that

4m — 3

if M > 1,

\a3 — Ma2 | = ^ 1 + 2exp[—2m/(1 — m)] if 0 < m< 1, 3 — 4m if M < 0.

(3)

The Hankel functional H2(1) = |a3 — a|| and H2(2) = |a2a4 — a§| is also known as Fekete-Szego functional and second Hankel determinant respectively. The Hankel functional has many applications in functional theory. For example \a3 — a2\ is equal to Sf(z)/6, where Sf(z) is the schwarzian derivative of the locally univalent function defined Sf(z) = (f''(z)/f'(z))' — 1/2(f''(z)/f (z))2 (See [19]). In 1969, Keough and Merkers [11] solved Fekete-Szego problem for the classes of starlike and convex functions. Lee et al. [13] established the sharp bounds to \H2(2)\ by generalizing several classes defined by subordination. Janteng et al. [9] (see also [1,18]) provided a brief survey on Hankel determinants and obtained bounds for \H2(2)\ for the classes of starlike and convex functions.

The theory of q-calculus in recent years has attracted the attention of researchers. The q-analogy of the ordinary derivative was initiated at the beginning of century by Jackson [8]. Ismail et al. [7] first introduce and explore class of generalized complex functions via q-calculus on the open unit disk U. Recently many newsworthy results related to subclass of analytic functions and q-operators are meticulously studied by various authors (see [10,17,20]). For 0 < q < 1, the q-derivative of a function f given by (1) is defined as

Dq f (z) =

f (qz) — f (z) (q — 1)z

J '(0)

for z = 0, for z = 0.

(4)

We note that lim Dqf (z) = f'(z). From (4), we deduce that

Dq f (z) = l + ^[fc]qak zk-1, (5)

k=2

where as q ^ l-

[k]q = -—^ = 1 + q + ••• + qk k. (6)

In this connection, our aim is to study upper bounds for functional \a2a4 — a3\ for functions belonging to the class f € S*(a), which is defined as follows.

Definition 0.1. A function f (z) given by (1) is said to be in the class f € S*(a), 0 < q < 1, 0 ^ a < 1 if the following conditions are satisfied:

f € Z(fZ((Z)) ^ (0 ^ 3< 1; z € U)

and z(DqgH) >/ (0 < /3< 1; z € U), (7)

g(w)

where the function g is the extension of f-1 to U.

In order to derive our main results, we have to recall here the following lemma.

Lemma 0.1 ([16]). If h € H, then \Bk\ ^ 2, for each k ^ 1 and the inequality is sharp for the

function —.

1—z

Lemma 0.2 ([6]). If p € P, p(z) = 1 + c1z + c2z2 + c3z3 + ... then 2c2 = cj + x(4 — of), 4c3 = cf + 2(4 — c2)c1x — c1 (4 — c1)x2 + 2(4 — c2)(1 — \x\2)z, for some x, z with \x\ < 1 and \z\ < 1.

Another result that will required is the maximum value of a quadratic expression. Stranded computation shows

max (Pt2 + Qt + R) = R if Q / 0, P / —Q/4, (8)

1. Main results

In this section, we investigate second Hankel determinant \H2(2)\ for functions belonging to the class S*(a) using q-differential operator. For convenience, in the sequel we use the abbreviation q2 = [2]q — 1, q3 = [3] q — 1, q4 = [4]q — 1.

Theorem 1.1. Let 0 ^ a < 1, 0 < q < 1. If function f € A given by (1) belongs to the class S*(a) then

i. For Q > 0, P < —Q/8

\a2a4 — a2U t(R — ^ . (9)

if Q > 0, "a /A -Q/8,

if Q /A 0, "a /A —Q/4,

if Q > 0, P > -Q/8 or Q A 0, P > —Q/4

2

ii. For Q < 0, P < —Q/4

iii. For Q > 0, P > —Q/4

\a2a4 — a|| ^ TR. (10)

|a2a4 — аЦ < T (16P + 4Q + R), (11)

where

P = 4ß2L + ßM + N, Q = U — 4ßV, R = 64q2q4, L =(q4 — q3)ql, M = ^4 + 8q3 — 8^4,

N = 4(q4 — qs) — qlq3q4 + 4q2,q 4, U = 4q^q3q4 + 12q^q^ — 32q^q4, V = q| qsq4 and (12) T =(1 — ß)2

4qmiq4

Proof. If f G S* (a) and g G f-1. Then

Zff)1= ß +(1 — ß)p(z)

and

We obtain

w(Dq g(w))

ß + (1 — ß)q(w). (13)

f(Z)

Also

g(w)

z(Dqf (z)) = 1 + q2a2Z + [q3a3 — q2a^] z2 + [q4a4 — (q2 + qs)asa2 + q2а%] z3 +----. (14)

w(Dq g(w)) [ ( 2 ) 2] 2

= 1 — q2a2z + q [2a^ — a3) — q2a.^\ w +

g(w)

+ [(q2 + q3)a2 (2a2 — a3) — q4 (5a2 — 5a2a3 + a^ — q2a\ From (13), (14) and (15), it is easily seen that

(1 — ß)ci (16)

a2 = -, (16)

q2

(1 — ß)2 c2i + (_l—ß)(c_2 — d2 )

q2 2qs

and

qs(1 — ß)3c3 , 5(1 — ß)2ci (c2 — d2) , (1 — ß)(cs — ds) (18)

a4 = -3--I----I--7,-. (18)

q'3 q4 4q2qs 2q4

Upon simplification, we easily establish

^ — ß) c1 . (1 — ß ) (c2 — d2 ) as = -12--1--7ÇZ--(17)

| 2| 1 a2 a4 — as1

(qs — q4)(1 — ß)4 4 , (1 — ß)3 2

q4q, ~c1 + c1 (c2 — d'2) +

^(cs — ds) — t1- 2

2q2q4 4q2

Ш4 Щ qs

+ ( d ) (1 — ß)2 ( d )2 + -7:-cic — ds)--—2— (c2 — d2)

According to Lemmas 1 and 2, we write

4 — c?

C2 — d2 = —-— (x — y)

(20)

and

c? , C2

4c3 — 4d3 = +

(4 — c?)

(x + y) —

c1

4 - c21

12

(x2 + y2) +

+ - ((- — Ix|2) z — (1 — |y|2) '

(21)

for some x,y,z and w with \x\ < 1, \y\ < 1, \z\ < 1 and \w\ < 1. Substituting values of c2,c3,d2 and d3 from (20), (21) on the right side of (19), we have

Ia2a4 — a31 A M? + M2(q? + 92) + M3(QÎ + Q2) + M4O + 92) := F(qi, Q2),

where

M1

M2

(q4 — q3)(- — ß)4 c4 , (- — ß)2 c4 , (- — ß)2 ci (4 c2)

-4-ci + :-ci + ö-ci I4 — cv ,

q|q4 4q2 q4 2q2q4

(1 — ß)3 c: (4 — cî) + Î-—^c? (4 — c?)

M3 =

8q|q3 (1 — ß)2

8q2q4

M4 =

(4 — c?) —

(1 — ß)2 8q2

4q2q4

(1 — ß)2 4q2q4

c? (4 — c2)

|x| + |y|),

(|x|2 + |y|2) ,

(4 — c2)2](|x| + |y|)2 .

Applying Lemma 1, without loss of generality assume c1 = c € [0, 2] for Q2 = \y\ ^ 1 and using triangle inequality, we have

M1 = [4 (q4 — q3) (1 — 3)2 — 2c3 + 8c + q23] c4 > 0,

4q2 q4

M2 = i1 — 3)2 [(1 — 3)q4 + 2q2q3] c2 (4 — c2) > 0, 8q2q3 q4

M3 = (1n — 3)2 (4 — c2) c(c — 2) < 0,

M4

8q2q4 = (1 — ß)2 4q3

(4 — c2)2 > 0.

(22)

(23)

(24)

(25)

(26)

x A 1 and

(27)

(28)

(29)

(30)

To maximize the function F(q1, q2) on the closed region & = {(g^ g2) : 0 < g1 < 1, 0 < g2 < 1}. Differentiating F(g1, g2) partially with respect to g1 and g2, we get

Feiei • Fß2ß2 (Feiß2) < 0

(31)

This shows that the function F(g1, g2) cannot have local maximum in the interior of the region &. Now we investigate the maximum of F(g1, g2) on the boundary of the region &. For g1 = 0 and 0 < g2 < 1 (similarly g2 = 0 and 0 < g1 < 1), we obtain

F (0, 92) = ^(02) = (M3 + M4) q2 + M2Q2 + M?.

(32)

2

?

i. M3 + M4 ^ 0 : In this case for 0 ^ g2 ^ 1 and any fixed c with 0 ^ c ^ 2 , it is clear that i'(g2) = 2 (M3 + M4) g2 + M2 > 0, that is i(g2) is an increasing function hence for fixed c G [0, 2), the maximum of i(g2) occurs at g2 = 1 and maximum of g2 = Mi + M2 + M3 + M4.

ii. M3 + M4 < 0 : Since M2 +2 (M3 + M4) > 0 for 0 < g2 < 1 and for any fixed c with 0 < c < 2, it is clear that M2 + 2 (M3 + M4) < 2 (M3 + M4) g2 + M2 < M2 and so i'(g2) > 0. Hence for fixed c with 0 ^ c < 2, the maximum i'(g2) occurs at g2 = 1. Also for c =2 we obtain

F (91,92)

4(1 - ß)2 (q4 - q3)

4

q2q4

(1 - ß)2 +

q3

(q4 - q3)_

For g1 = 1 and 0 < g2 < 1 (similarly g2 = 1 and 0 < g1 < 1), we obtain

F (1, g2) = U (g2) = (M3 + M4) 92 + (M2 + 2M4) 92 + Mi + M2 + M3 + M4. Thus from above cases of M3 + M4 we get that

max U (g2) = U (1) = M1 + 2M2 + 2M3 + 4M4.

(33)

(34)

(35)

Since i(1) < U(1) for c G [0,2], we obtain max F (g1,g2) = F(1,1) on the boundary of the square 6. Thus , the maximum of F occurs at g1 = 1 and g2 = 1 in the closed square 6. Let k : [0,2] —> R defined by

k(c) = max (g1: g2) = F(1,1) = M1 + 2M2 + 2M3 + 4M4. (36)

Substituting the values of M^ M2, M3, M4 in the function k defined by (36), we get

(c)

(1 -ß)2 4q2qlq4

(|4 (q4 - qs) (1 - ß?q3 - (1 - ß)q2q3q4 + 4q^q^^c4+ + 14(1 - ß)qlqsq4 + 1244 - 32q^q4^c2 + |64q4q4^

(37)

which is quadratic in c2. Using the standard computation, we get

(4PR - Q2)/4P if Q> 0, P < -Q/8, R if Q < 0, P < -Q/4,

16P + 4Q + R if Q > 0, P > -Q/8 or Q < 0, P > -Q/4

where P, Q, R and T are given by (12). This completes the proof.

Theorem 1.2. Let 0 < q < 1, 0 < a < 1 and f G S*(a). Then for complex ^

, 21. (2 - V)(1 - P)2

I03 — ua2 I ^ -2-.

q22

Proof. Letting c := c1 > 0. Then for complex (1, using (16) and (17), we have

□

(39)

2 (1 - ß f c2 + (±-ß)(c2-d2l (1 - ß)2 c a3 - (1,0,2 = -2--+--ö--(

22

q22

2q3

q22

(2 - ()(1 - ß)2c2qs + (1 - ß)(c2 - d2) 2q2q3 '

By (16), we obtain

2 _ 2(2 - M)(1 - ß)2c2q3 + (1 - ß)(4 - c2)(x - y) %2 —

where x and y satisfying \x\ < 1, \y\ < 1

«3 - = -—2-, (41)

2, ^ (2 - M)(1 - ß)2c2

I «3 — 1

using c < 2, we get

«3 - M«2| < -T"2-, (42)

4?2

, 2|. (2 - M)(1 ~ 3)2 (43)

|a3 — ^a2| ^ -2-• (43)

q2.

This completes the proof. □

I am grateful to the reviewer(s) of this article who gave valuable suggestions in order to improve and revise the paper in present form.

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Второй определитель Ганкеля для биунивалентных функций, ассоциированных с ^-дифференциальным оператором

Малликарджун Г. Шриган

Технологический и исследовательский институт Бхиварабаи Саванта

Пуна, штат Махараштра, Индия

Аннотация. Целью данной статьи является получение верхней оценки второго определителя Ганкеля, обозначаемого Н2(2), для класса Я* (а) биунивалентных функций используя д-дифференциальный оператор.

Ключевые слова: определитель Ганкеля, биоднолистные функции, д-дифференциальный оператор, функционал Фекете-Сегао.