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Second Hankel Determinants for Some Subclasses of Biunivalent Functions Associated with Pseudo-Starlike Functions.

1. Introduction

Let A be the class of all analytic functions f of the form

f(z) = z + [[infinity].summation over (n=2)][a.sub.n][z.sup.n], (1)

in the open unit disc [DELTA] = {z :[absolute value of z] < 1}. Let S be the subclass of A consisting of univalent functions. Let P be the family of analytic functions p(z) in [DELTA] such that p(0) = 1 and [R.sub.p](z) > 0(z [member of] [DELTA]) of the form

p(Z) = 1 + [[infinity].summation over (n=1)][c.sub.n][z.sup.n]. (2)

For any two functions f and g analytic in [DELTA], we say that the function f is subordinate to g in [DELTA] and we write it as f(z) [??] g(z),if there exists an analytic function w, in [DELTA] with w(0) = 0, [absolute value of (w(z))] < 1 (z [member of] [DELTA]) such that f(z) = g(w(z)). In view of Koebe 1/4 theorem, every function f [member of] S has an inverse [f.sup.-1], defined by

[f.sup.-1](f(z)) = z, (z [member of] [DELTA]),

f([f.sup.-1](w)) = w ([absolute value of w] < [r.sub.0](f); [r.sub.0](f) [greater than or equal to] [1/4]). (3)

In fact, the inverse function is given by

[f.sup.-1](w) = w - [a.sub.2][w.sup.2] + (2[a.sup.2.sub.2] - [a.sub.3])[w.sup.3] - (5[a.sup.3.sub.2] - 5[a.sub.2][a.sub.3] + [a.sub.4]) [w.sup.4] + .... (4)

A function f [member of] A is said to be biunivalent in [DELTA] if both f and [f.sup.-1] are univalent in [DELTA]. Let [SIGMA] denote the class of all biunivalent functions defined in the unit disc [DELTA]. We notice that [SIGMA] is nonempty. The behavior of the coefficients is unpredictable when the biunivalency condition is imposed on the function f [member of] A.In 1967, Lewin [1] introduced the class [SIGMA] of biunivalent functions and investigated second coefficient in Taylor-Maclaurin series expansion for every f [member of] [SIGMA]. Subsequently, in 1967, Brannan and Clunie [2] introduced bistarlike functions and biconvex functions similar to the familiar subclasses of univalent functions consisting of strongly starlike, strongly convex, starlike, and convex functions and so on and obtained estimates on the initial coefficients conjectured that [absolute value of ([a.sub.2])] [less than or equal to] [square root of 2] for bistarlike functions and [absolute value of ([a.sub.2])] [less than or equal to] 1 for biconvex functions. Only the last estimate is sharp; equality occurs only for f(z) = z/(1 - z) or its rotation. Since then, various subclasses of biunivalent functions class Z were introduced and nonsharp estimates on the first two coefficients [absolute value of ([a.sub.2])] and [absolute value of ([a.sub.3])] in Taylor-Maclaurin series expansion were found in several investigations. The coefficient estimate problem for each of [absolute value of ([a.sub.n])] is still an open problem. In 1976, Noonan and Thomas [3] defined qth Hankel determinants of f for q [greater than or equal to] 1 and n [greater than or equal to] 1 which is stated as follows:

[mathematical expression not reproducible] (5)

Easily one can observe that [H.sub.2](1) = [absolute value of ([a.sub.3] - [a.sup.2.sub.2])] is a special case of the well known Fekete-Szego functional [absolute value of ([a.sub.3] - [mu][a.sup.2.sub.2])] where [mu] is real, for [mu] = 1. Now for q = 2, n = 2, we get second Hankel determinant

[mathematical expression not reproducible] (6)

In particular, sharp upper bounds on [H.sub.2](2) were obtained by the authors of articles [4-6] for various subclasses of analytic and univalent functions. In 2013, Babalola [7] determined the second Hankel determinant with Fekete-Szego parameter [absolute value of ([a.sub.2][a.sub.4] - [lambda][a.sup.2.sub.3])] for some subclasses of analytic functions. Let [phi] be an analytic function with positive real part in [DELTA] such that [phi](0) = 1, [phi]'(0) > 0 which is symmetric with respect to the real axis. Such a function has a Maclaurin series expansion of the form [phi](z) = 1 + [B.sub.1]z + [B.sub.2][z.sup.2] + [B.sub.3][z.sup.3] + ... ([B.sub.1] > 0).

Researchers like Duren [8], Singh [9], and so on have studied various subclasses of usual known Bazilevic function of order a denoted by B([alpha]) which satisfy the geometric condition Re(f[(z).sup.[alpha]-1] f'(z)/[z.sup.[alpha]-1)] > 0, where [alpha] is nonnegative real number, different ways of perspectives of convexity, radii of convexity and starlikeness, inclusion properties, and so on. The class B([alpha]) reduces to the starlike function and bounded turning function whenever [alpha] = 0 and [alpha] = 1, respectively. This class is extended to B([alpha], [beta]) which satisfy the geometric condition Re(f[(z).sup.[alpha]-1] f'(z)/[z.sup.[alpha]-1]) > [beta], where [alpha] is nonnegative real number and 0 [less than or equal to] [beta] < 1. Recently, Babalola [7] defined new subclass [lambda]-pseudo-starlike functions of order [beta] (0 [less than or equal to] [beta] < 1) which satisfy the condition Re(z[[f'(z)].sup.[lambda]]/f(z)) > [beta], ([lambda] [greater than or equal to] 1 [member of] R, 0 [less than or equal to] [beta] < 1, z [member of] [DELTA]) and is denoted by [L.sub.[lambda]]([beta]). Babalola [7] proved that all pseudo-starlike functions are Bazilevic of type (1 - 1/[lambda]), order [[beta].sup.(1/[lambda]], and univalent in the open unit disc [DELTA]. For [lambda] = 2 we note that functions in [L.sub.2]([beta]) are defined by Re f'(z)(zf'(z)/f(z)) > [beta] which is a product combination of geometric expressions for bounded turning and starlike functions. Note that the singleton subclass [L.sub.[infinity]]([beta]) of S contains the identity map. In 2016, Joshi et al. [10] defined two new subclasses of biunivalent functions using pseudo-starlike functions, one is L[B.sup.[lambda].sub.[SIGMA]]([alpha]) class of strong [lambda]-bi-pseudo-starlike functions of order [alpha] and other is L[B.sub.[SIGMA]]([lambda], [beta]) [lambda]-bi-pseudo-starlike functions of order [beta] in the open unit disc. Many researchers [11-15] have estimated the second Hankel determinants for some subclasses of biunivalent functions. Motivated by the above-mentioned work, in this paper we have introduced [lambda]-bi-pseudo-starlike functions subordinate to a starlike univalent function whose range is symmetric with respect to the real axis and estimated second Hankel determinants.

Definition 1. A function f [member of] [SIGMA] is said to be in the class L[B.sup.[lambda].sub.[SIGMA]]([phi]), [lambda] [greater than or equal to] 1, if it satisfies the following conditions:

z[[f'(z)].sup.[lambda]/f(z)] [??] [phi](z), z [member of] [DELTA],

w[[g'(w)].sup.[lambda]/g(w)] [??] [phi](w), w [member of] [DELTA], (7)

where g is an extension of [f.sup.-1] to [DELTA].

(1) If [phi](z) = (1 + z)/(1 - z), then the class L[B.sup.[lambda].sub.[SIGMA]]([phi]) reduces to the class L[B.sup.[lambda].sub.[SIGMA]] and satisfies the following conditions:

RE[z[[f'(z)].sup.[lambda]]/f(z)] > 0, z [member of] [DELTA],

Re[w[[g'(w)].sup.[lambda]]/g(w)] > 0, w [member of] [DELTA], (8)

where g is an extension of [f.sup.-1] to [DELTA].

(2) If [phi](z) = (1 + (1 - 2[alpha])z)/(1 - z), 0 [less than or equal to] [alpha] < 1, then the class L[B.sup.[lambda].sub.[SIGMA]]([phi]) reduces to the class L[B.sup.[lambda].sub.[SIGMA]]([alpha]) and satisfies the following conditions:

RE[z[[f'(z)].sup.[lambda]]/f(z)] > [alpha], z [member of] [DELTA],

Re[w[[g'(w)].sup.[lambda]]/g(w)] > [alpha], w [member of] [DELTA], (9)

where g is an extension of [f.sup.-1] to [DELTA].

(3) If [phi](z) = [((1 + z)/(1 - z)).sup.[beta]], then the class L[B.sup.[lambda].sub.[SIGMA]] ([phi]) reduces to the class L[B.sup.[lambda].sub.[SIGMA]]([beta]), 0 < [beta] [less than or equal to] 1 and satisfies following conditions:

[mathematical expression not reproducible] (10)

where g is an extension of [f.sup.-1] to [DELTA].

(4) If [lambda] = 1, then the class L[B.sup.[lambda].sub.[SIGMA]]([phi]) reduces to the class of bistarlike functions S[T.sub.[SIGMA]]([phi]) and satisfies the following conditions:

zf'(z)/f(z) [??] [phi](z), z [member of] [DELTA],

wg'(w)/g(w) [??] [phi](w), w [member of] [DELTA], (11)

where g is an extension of [f.sup.-1] to [DELTA].

Several choices of [phi] would reduce the class S[T.sub.[SIGMA]]([phi]) to some well known subclasses of [SIGMA].

(1) For the function [phi] given by [phi](z) = (1 + (1 - 2[alpha])z)/(1 - z), 0 [less than or equal to] [alpha] < 1, the class S[T.sub.[SIGMA]]([phi]) reduces to the class S[T.sub.[SIGMA]]([alpha]) and satisfies the following conditions:

RE[zf'(z)/f(z)] > [alpha], z [member of] [DELTA],

Re[wg'(w)/g(w)] > [alpha], w [member of] [DELTA], (12)

where g is an extension of [f.sup.-1] to [DELTA] and this class is called class of bistarlike function of order [alpha].

(2) For the function [phi] given by [phi](z) = (1 + z)/(1 - z), the class S[T.sub.[SIGMA]]([phi]) reduces to the class S[T.sub.[SIGMA]] and satisfies the following conditions:

RE[zf'(z)/f(z)] > 0, z [member of] [DELTA],

Re[wg'(w)/g(w)] > 0, w [member of] [DELTA], (13)

where g is an extension of [f.sup.-1] to [DELTA] and this class is called class of bistarlike function.

2. Preliminary Lemmas

Let P denote the class of functions consisting of p, such that

p(z) = 1 + [c.sub.1]z + [c.sub.2][z.sup.2] + [c.sub.3][z.sup.3] + ... = 1 + [[infinity].summation over (n=1)][c.sub.n][z.sup.n] (14)

which are analytic in the open unit disc [DELTA] and satisfy R{p(z)} > 0 for any z [member of] [DELTA].

Lemma 2 (see [8]). If p [member of] P, then [absolute value of ([c.sub.n])] [less than or equal to] 2 for each n [greater than or equal to] 1 and the inequality is sharp for the function (1 + z)/(1 - z).

Lemma 3 (see [16]). The power series for p(z) = 1 + [[summation].sup.[infinity].sub.n=1][c.sub.n][z.sup.n] given in (14) converges in the open unit disc [DELTA] to a function in P if and only if the Toeplitz determinants

[mathematical expression not reproducible] (15)

and [c.sub.-k] = [bar.[c.sub.k]] are all nonnegative. They are strictly positive except for p(z) = [[summation].sup.m.sub.k=1][[rho].sub.k][p.sub.o](exp([it.sub.k])z), [[rho].sub.k] > 0, [t.sub.k] real, and [t.sub.k] [not equal to] [t.sub.j], for k [not equal to] j, where [p.sub.o](z) = (1 + z)/(1 - z); in this case [D.sub.n] > 0 for n < (m - 1) and [D.sub.n] = 0 for n [greater than or equal to] m.

We may assume without any restriction that [c.sub.1] > 0, on using Lemma 3 for n = 2 and n = 3, respectively, we have

[mathematical expression not reproducible] (16)

which is equivalent to

2[c.sub.2] = [c.sup.2.sub.1] + x(4 - [c.sup.2.sub.1], for some x, [absolute value of x] [less than or equal to] 1. (17)

If we consider the determinant

[mathematical expression not reproducible] (18)

we get the following inequality:

[absolute value of ([4[c.sub.3] - 4[c.sub.1][c.sub.2] + [c.sup.3.sub.1])(4 - [c.sup.2.sub.1]) + [c.sub.1] [(2[c.sub.2] - [c.sup.2.sub.1]).sup.2])] [less than or equal to] 2[(4 - [c.sup.2.sub.1]).sup.2] - 2 [[absolute value of ((2[c.sub.2] - [c.sup.2.sub.1]))].sup.2]. (19)

From (17) and (19), it is obtained that

4[c.sub.3] = [c.sup.3.sub.1] + 2[c.sub.1] (4 - [c.sup.2.sub.1]) x - [c.sub.1] (4 - [c.sup.2.sub.1]) [x.sup.2] + 2[c.sub.1](4 - [c.sup.2.sub.1])(1 - [[absolute value of x].sup.2])z (20)

for some z, [absolute value of z] [less than or equal to] 1.

Another required result is the optimal value of quadratic expression. Standard computations show that

[mathematical expression not reproducible] (21)

3. Main Results

Theorem 4. If f [member of] L[B.sup.[lambda].sub.[SIGMA]]([phi]) and is of the form (1) then we have the following.

(1) [absolute value of ([a.sub.2])] [less than or equal to] [B.sub.1]/(2[lambda] - 1).

(2) [absolute value of ([a.sub.3])] [less than or equal to] [B.sup.2.sub.1]/[(2[lambda] - 1).sup.2] + [B.sub.1]/(3[lambda] - 1).

(3) [absolute value of ([a.sub.4])] [less than or equal to] [B.sup.3.sub.1][absolute value of ((-4[[lambda].sup.3] + 13[lambda] - 3))]/3[(2[lambda] - 1).sup.3](4[lambda] - 1) + 5[B.sup.2.sub.1]/2(2[lambda] - 1)(3[lambda] - 1) + 4[B.sub.1]/(4[lambda] - 1) + [absolute value of ([B.sub.3])]/(4[lambda] - 1).

Proof. Since f [member of] L[B.sup.[lambda].sub.[SIGMA]]([phi]), there exist two Schwartz functions u(z), v(w) in [DELTA] with u(0) = 0, v(0) = 0 and [absolute value of (u(z))] [less than or equal to] 1, [absolute value of (v(w))] [less than or equal to] 1 such that

[z[[f'(z)].sup.[lambda]]/f(z)] = {[phi][u(z)]},

[w[[g'(z)].sup.[lambda]]/g(w)] = {[phi][v(w)]}. (22)

Define two functions p(z), q(w) such that

p(z) = [1 + u(z)/1 - u(z)] = 1 + [c.sub.1]z + [c.sub.2][z.sup.2] + [c.sub.3][z.sup.3] + ...,

q(w) = [1 + v(w)/1 - v(w)] = 1 + [d.sub.1]w + [d.sub.2][w.sup.2] + [d.sub.3][w.sup.3] + .... (23)

Then

[mathematical expression not reproducible] (24)

Then (22) becomes

z[[f'(z)].sup.[lambda]]/f(z) = [phi] (1 + u(z)/1 - u(z)),

w[[g'(w)].sup.[lambda]]/g(w) = [phi] (1 + v(w)/1 - v(w)). (25)

Now equating the coefficients in (25)

(2[lambda] - 1)[a.sub.2] = [B.sub.1][c.sub.1]/2, (26)

(3[lambda] - 1)[a.sub.3] - (-2[[lambda].sup.2] + 4[lambda] - 1)[a.sup.2.sub.2] = [[B.sub.1]/2]([c.sub.2] - [[c.sup.2.sub.1]/2]) + [[B.sub.2][c.sup.2.sub.1]/4], (27)

[mathematical expression not reproducible] (28)

-(2[lambda] - 1)[a.sub.2] = [[B.sub.1][d.sub.1]/2], (29)

(2[[lambda].sup.2] + 2[lambda] - 1)[a.sup.2.sub.2] - (3[lambda] - 1)[a.sub.3] = [[B.sub.1]/2]([d.sub.2] - [[d.sup.2.sub.1]/2]) + [[B.sub.2][d.sup.2.sub.1]/4], (30)

[mathematical expression not reproducible] (31)

Now from (26) and (29)

[c.sub.1] = -[d.sub.1],

[a.sub.2] = [B.sub.1][c.sub.1]/2(2[lambda] - 1). (32)

Now from (27) and (30)

[a.sub.3] = [[B.sup.2.sub.1][c.sup.2.sub.1]/4[(2[lambda] - 1).sup.2]] + [[B.sub.1]([c.sub.2] - [d.sub.2])/4(3[lambda] - 1)]. (33)

Now from (28) and (31)

[mathematical expression not reproducible] (34)

and with the help of the above Lemma 2, we get the required results.

Theorem 5. If L[B.sup.[lambda].sub.[SIGMA]]([phi]) is of the form (1) then

[mathematical expression not reproducible] (35)

Proof. Now adding (27) and (30), we get that

(4[[lambda].sup.2] - 2[lambda])[a.sup.2.sub.2] = [[B.sub.1]]([c.sub.2] + [d.sub.2]) + ([c.sup.2.sub.1] + [d.sup.2.sub.1])[[B.sub.2] - [B.sub.1]/4]. (36)

Now from (26) and (29), we get that

[a.sup.2.sub.2] = [B.sup.2.sub.1]([c.sup.2.sub.1] + [d.sup.2.sub.1])/8[(2[lambda] - 1).sup.2]. (37)

Now from (36) and (37)

[absolute value of ([a.sub.3] - [mu][a.sup.2.sub.2])]

= [[B.sub.1]/4(3[lambda] - 1)] [[c.sub.2](1 + h([mu])) + [d.sub.2] (-1 + h([mu]))], (38)

where h([mu]) = [B.sup.2.sub.1](1 - [mu])(3[lambda] - 1)/[[lambda](2[lambda] - 1)[B.sup.2.sub.1] + ([B.sub.1] - [B.sub.2])[(2[lambda] - 1).sup.2]] which completes the proof of the theorem.

Theorem 6. If L[B.sup.[lambda].sub.[SIGMA]]([phi]) is of the form (1) then we have the following.

(1) If 4[[xi].sub.1] [less than or equal to] [[xi].sub.3], [[xi].sub.2] [less than or equal to] [B.sub.1]/2[(3[lambda] - 2).sup.2] then [absolute value of ([a.sub.2][a.sub.4] - [a.sup.2.sub.3])] [less than or equal to] [B.sup.2.sub.1]/[(3[lambda] - 1).sup.2].

(2) If 4[[xi].sub.1] [greater than or equal to] [[xi].sub.3], [[xi].sub.1] - [[xi].sub.2]/2 - [B.sub.1](1/(2[lambda] - 1)(4[lambda] - 1) - 1/[(3[lambda] - 1).sup.2]) [greater than or equal to] 0 or 4[[xi].sub.1] [less than or equal to] [[xi].sub.3], [[xi].sub.2] [greater than or equal to] [B.sub.1]/2[(3[lambda] - 2).sup.2] then [absolute value of ([a.sub.2][a.sub.4] - [a.sup.2.sub.3])] [less than or equal to] 2[B.sub.1][[xi].sub.2].

(3) 4[[xi].sub.1] > [[xi].sub.3], [[xi].sub.1] - [[xi].sub.2]/2 - [B.sub.1](1/(2[lambda] - 1)(4[lambda] - 1) - 1/[(3[lambda] - 1).sup.2]) [less than or equal to] 0; then

[mathematical expression not reproducible] (39)

where

[mathematical expression not reproducible] (40)

Proof. Using the values of [a.sub.2], [a.sub.3], [a.sub.4] from the above theorem, one can obtain

[mathematical expression not reproducible] (41)

According to Lemma 3 we get that

[mathematical expression not reproducible] (42)

For some z, w with [absolute value of z] [less than or equal to] 1, [absolute value of w] [less than or equal to] 1. using (42), we have

[mathematical expression not reproducible] (43)

Since p [micro] P, [absolute value of ([c.sub.1])] [less than or equal to] 2. Letting [c.sub.1] = c we may assume without any restriction that c [member of] [0,2]. Thus for [[gamma].sub.1] = [absolute value of x] [less than or equal to] 1 and [[gamma].sub.2] = [absolute value of y] [less than or equal to] 1, we obtain

[absolute value of ([a.sub.2][a.sub.4] - [a.sup.2.sub.3])] [less than or equal to] [T.sub.1] + [T.sub.2]([[gamma].sub.1] + [[gamma].sub.2]) + [T.sub.3]([[gamma].sup.2.sub.1] + [[gamma].sup.2.sub.2]) + [T.sub.4][([[gamma].sub.1] + [[gamma].sub.2]).sup.2] = F([[gamma].sub.1], [[gamma].sub.2]), (44)

where

[mathematical expression not reproducible] (45)

Now we need to maximize F([[gamma].sub.1], [[gamma].sub.2]) in the closed square S = [0,1] x [0,1] for c [member of] [0,2]. We must investigate the maximum of F([[gamma].sub.1], [[gamma].sub.2]) according to c [member of] (0,2), c = 2, and c = 0 taking into account the sign of [mathematical expression not reproducible].

First, let c [member of] (0,2). Since [T.sub.3] < 0 and [T.sub.3] + 2[T.sub.4] > 0, we conclude that [mathematical expression not reproducible]. Thus the function F cannot have a local maximum in the interior of the square S. Now, we investigate the maximum of F on the boundary of the square S.

For [[gamma].sub.1] = 0 and 0 [less than or equal to] [[gamma].sub.2] [less than or equal to] 1 (similarly [[gamma].sub.2] = 0 and 0 [less than or equal to] [[gamma].sub.1] [less than or equal to] 1), we obtain

F(0, [[gamma].sub.2]) = G ([[gamma].sub.2]) = [T.sub.1] + [T.sub.2][[gamma].sub.2] + ([T.sub.3] + [T.sub.4]) [[gamma].sup.2.sub.2]. (46)

(i) The Case [T.sub.3] + [T.sub.4] [greater than or equal to] 0. In this case 0 [less than or equal to] [[gamma].sub.2] [less than or equal to] 1 and for any fixed c with 0 < c < 2, it is clear that G'([[gamma].sub.2]) = 2([T.sub.3] + [T.sub.4])[[gamma].sub.2] + [T.sub.2] > 0; that is, G([[gamma].sub.2]) is an increasing function. Hence for any fixed c [member of] (0,2) the maximum of G([[gamma].sub.2]) occurs at [[gamma].sub.2] = 1 and

max G ([[gamma].sub.2]) = G(1) = [T.sub.1] + [T.sub.2] + [T.sub.3] + [T.sub.4]. (47)

(ii) The Case [T.sub.3] + [T.sub.4] < 0. Since 2([T.sub.3] + [T.sub.4]) + [T.sub.2] [greater than or equal to] 0 for 0 [less than or equal to] [[gamma].sub.2] [less than or equal to] 1 and for any fixed c with 0 < c < 2, it is clear that 2([T.sub.3] + [T.sub.4]) + [T.sub.2] < 2([T.sub.3] + [T.sub.4])[[gamma].sub.2] + [T.sub.2] < [T.sub.2] and so G'([[gamma].sub.2]) > 0. Hence for any fixed c [member of] [0,2) the maximum of G([[gamma].sub.2]) occurs at [[gamma].sub.2] = 1. Also for c = 2 we obtain

F([[gamma].sub.1], [[gamma].sub.2]) = [B.sub.1]([[lambda](2[lambda] + 1)[B.sup.3.sub.1]/3[(2[lambda] - 1).sup.3](4[lambda] - 1)] + [[absolute value of ([B.sub.3])]/(2[lambda] - 1)(4[lambda] - 1))]. (48)

Taking into account the value of (48) and case (i) and case (ii), for 0 [less than or equal to] [[gamma].sub.2] [less than or equal to] 1 and for any fixed c with 0 [less than or equal to] c [less than or equal to] 2,

maxG([[gamma].sub.2])= G(1) = [T.sub.1] + [T.sub.2] + [T.sub.3] + [T.sub.4]. (49)

For [[gamma].sub.1] = 1 and 0 [less than or equal to] [[gamma].sub.2] [less than or equal to] 1 (similarly [[gamma].sub.2] = 1 and 0 [less than or equal to] [[gamma].sub.1] [less than or equal to] 1), we obtain

F(1, [[gamma].sub.2]) = H([[gamma].sub.2]) =([T.sub.3] + [T.sub.3])[[gamma].sup.2.sub.2] + ([T.sub.2] + 2[T.sub.4])[[gamma].sub.2] + [T.sub.1] + [T.sub.2] + [T.sub.3] + [T.sub.4]. (50)

Similar to the above case of [T.sub.3] + [T.sub.4], we get that

max H([[gamma].sub.2]) = H(1) = [T.sub.1] + 2[T.sub.2] + 2[T.sub.3] + 4[T.sub.4]. (51)

Since G(1) [less than or equal to] H(1) for c [member of] [0,2], max F([[gamma].sub.1], [[gamma].sub.2]) = F(1,1) on the boundary of the square S. Thus the maximum of F occurs at [[gamma].sub.1] = 1 and [[gamma].sub.2] = 1 in the closed square S.

Letting K : [0,2] [right arrow] R,

K(c) = max F([[gamma].sub.1], [[gamma].sub.2]) = F(1,1) = [T.sub.1] + 2[T.sub.2] + 2[T.sub.3] + 4[T.sub.4]. (52)

Substituting the values of [T.sub.1], [T.sub.2], [T.sub.3], [T.sub.4] in the above equation,

[mathematical expression not reproducible] (53)

Let

[mathematical expression not reproducible] (54)

Then K(c) = P[t.sup.2] + Qt + R, where t = [c.sup.2].

Then with help of optimal value of quadratic expression, we get the required result. This completes the proof of the theorem.

Corollary 7. If f [member of] L[B.sup.[lambda].sub.[SIGMA]] and is of the form (1) then

[absolute value of ([a.sub.2][a.sub.4] - [a.sup.2.sub.3])] [less than or equal to] 4[[4[lambda](2[lambda] + 1)/3[(2[lambda] - 1).sup.3](4[lambda] - 1)] + [1/(2[lambda] - 1)(4[lambda] - 1)]]. (55)

Corollary 8. If f [member of] L[B.sup.[lambda].sub.[SIGMA]]([alpha]) and is of the form (1) then

[mathematical expression not reproducible] (56)

where

[mathematical expression not reproducible] (57)

Corollary 9. If f [member of] L[B.sup.[lambda].sub.[SIGMA]]([beta]) and is of the form (1) then

[mathematical expression not reproducible] (58)

where

[mathematical expression not reproducible] (59)

Corollary 10. If f [member of] S[T.sub.[SIGMA]]([phi]) and is of the form (1) then we have the following.

(1) If 4([B.sup.2.sub.1]/8 + [absolute value of ([B.sub.2])]/3) [less than or equal to] [B.sub.1]/3, ([B.sup.3.sub.1] + [absolute value of ([B.sub.3])]/6 [less than or equal to] [B.sub.1]/8 then [absolute value of ([a.sub.2][a.sub.4] - [a.sup.2.sub.3])] [less than or equal to] [B.sup.2.sub.1]/4.

(2) If 4([B.sup.2.sub.1]/8 + [absolute value of ([B.sub.2])]/3) [greater than or equal to] [B.sub.1]/3, ([B.sup.3.sub.1] + [absolute value of ([B.sub.3])])/6 - (l/2)([B.sup.2.sub.1]/8 + [absolute value of ([B.sub.2])]/3) - [B.sub.1]/12 [greater than or equal to] 0 or 4([B.sup.2.sub.1]/8 + [absolute value of ([B.sub.2])]/3) [less than or equal to] [B.sub.1]/3, ([B.sup.3.sub.1] + [absolute value of ([B.sub.3])])/6 [greater than or equal to] [B.sub.1]/8 then [absolute value of ([a.sub.2][a.sub.4] - [a.sup.2.sub.3])] [less than or equal to] [B.sub.1]([B.sup.3.sub.1] + [absolute value of ([B.sub.3])])/3.

(3) 4([B.sup.2.sub.1]/8 + [absolute value of ([B.sub.2])]/3) > [B.sub.1]/3, ([B.sup.3.sub.1] + [absolute value of ([B.sub.3])])/6 - (1/2)([B.sup.2.sub.1]/8 + [absolute value of ([B.sub.2])]/3) - [B.sub.1]/12 [less than or equal to] 0; then

[mathematical expression not reproducible] (60)

The above result is obtained by taking [lambda] = 1 in Theorem 6, which is the second Hankel determinant of bistarlike function.

Corollary 11. If f [member of] S[T.sub.[SIGMA]]([alpha]) and is of the form (1) then

[mathematical expression not reproducible] (61)

Corollary 12. If f [member of] S[T.sub.[SIGMA]] and is of the form (1) then

[absolute value of ([a.sub.2][a.sub.4] - [a.sup.2.sub.3])] [less than or equal to] [20/3]. (62)

https://doi.org/10.1155/2017/6476391

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

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K. Rajya Laxmi (1,2) and R. Bharavi Sharma (1,2)

(1) Department of Mathematics, Kakatiya University, Telangana 506009, India

(2) Department of Mathematics, SR International Institute of Technology, Hyderabad, Telangana 501301, India

Correspondence should be addressed to K. Rajya Laxmi; rajyalaxmi2206@gmail.com

Received 25 August 2017; Accepted 7 November 2017; Published 4 December 2017

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Title Annotation:Research Article
Author:Laxmi, K. Rajya; Sharma, R. Bharavi
Publication:Journal of Complex Analysis
Article Type:Report
Date:Jan 1, 2017
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