# Coefficient Bounds for Subclasses of Biunivalent Functions Associated with the Chebyshev Polynomials.

1. Introduction

Let A denote the class of analytic functions of the form

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

normalized by the conditions f(0) = 0 = f'(0) - 1 defined in the open unit disk

[DELTA] = {z [member of] C : [absolute value of z] < 1}. (2)

Let S be the subclass of A consisting of functions of form (1) which are also univalent in [DELTA]. Let [S.sup.*]([alpha]) and K([alpha]) denote the well-known subclasses of S, consisting of starlike and convex functions of order [alpha] (0 [less than or equal to] [alpha] < 1), respectively.

The Koebe one-quarter theorem  ensures that the image of [DELTA] under every univalent function f [member of] A contains a disk of radius 1/4. Thus every univalent function f has an inverse [f.sup.-1] satisfying

[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)

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 biunivalent functions defined in the unit disk [DELTA]. Since f [member of] [SIGMA] has the Maclaurin series given by (1), a computation shows that its inverse g = [f.sup.-1] has the expansion

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

An analytic function f is subordinate to an analytic function g, written as f(z) [??] g(z), provided there is an analytic function w defined on [DELTA] with w(0) = 0 and [absolute value of (w(z))] < 1 satisfying f(z) = g(w(z)).

Chebyshev polynomials, which are used by us in this paper, play a considerable role in numerical analysis. We know that the Chebyshev polynomials are four kinds. The most of books and research articles related to specific orthogonal polynomials of Chebyshev family contain essentially results of Chebyshev polynomials of first and second kinds [T.sub.n](x) and [U.sub.n](x) and their numerous uses in different applications; see Doha  and Mason .

The well-known kinds of the Chebyshev polynomials are the first and second kinds. In the case of real variable x on (-1,1), the first and second kinds are defined by

[T.sub.n](x) = cos n[theta],

[U.sub.n](x), sin(n + 1)[theta]/sin [theta], (5)

where the subscript n denotes the polynomial degree and x = cos [theta]. We consider the function

[PHI](z, t) = 1/1 - 2tz + [z.sup.2]. (6)

We note that if t = cos [alpha], [alpha] [member of] (-[pi]/3, [pi]/3), then for all z [member of] [DELTA]

[PHI](z, t) = [1/1 - 2tz + [z.sup.2]] = 1 + [[infinity].summation over (n=1)][sin(n + 1)[alpha]/sin [alpha]] [z.sup.n] = 1 + 2 cos [alpha]z + (3 [cos.sup.2][alpha] - [sin.sup.2][alpha]) [z.sup.2] + .... (7)

Thus, we write

[PHI](z,t) = 1 + [U.sub.1](t)z + [U.sub.2](t)[z.sup.2] + ... (z [member of] [DELTA], t [member of] (-1, 1)), (8)

where [U.sub.n-1] = sin(n arccos f)/[square root of (1 - [t.sup.2])], for n [member of] N, are the second kind of the Chebyshev polynomials. Also, it is known that

[U.sub.n](t) = 2t[U.sub.n-1](t) - [U.sub.n-2](t),

[U.sub.1](t) = 2t; (9)

[U.sub.2](t) = 4[t.sup.2] - 1,

[U.sub.3] (t) = 8[t.sup.3] -4t, ... (10)

The Chebyshev polynomials [T.sub.n](t), t [member of] [-1,1], of the first kind have the generating function of the form

[[infinity].summation over (n=0)][T.sub.n](t)[z.sup.n] = [1 - tz/1 - 2tx + [z.sup.2]] (z [member of] [DELTA]). (11)

All the same, the Chebyshev polynomials of the first kind [T.sub.n](t) and the second kind [U.sub.n](t) are well connected by the following relationship:

[mathematical expression not reproducible] (12)

Several authors have introduced and investigated subclasses of biunivalent functions and obtained bounds for the initial coefficients (see [4-10]). In , making use of the Salagean  differential operator,

[D.sup.k] : A [right arrow] A (13)

defined by

[mathematical expression not reproducible] (14)

[D.sup.k]f(z) = z + [[infinity].summation over (n=2)][n.sup.k][a.sub.n][z.sup.n], k [member of] [N.sub.0] = N [union] {0}, (15)

and further for functions g of the form (4) Vijaya et al.  (also see ) defined

[D.sup.k]g(w) = w - [a.sub.2][2.sup.k][w.sup.2] + (2[a.sup.2.sub.2] - [a.sub.3]) [3.sup.k][w.sup.3] + ... (16)

and introduced two new subclasses of biunivalent functions. In this paper, we use Chebyshev polynomials to obtain the estimates on the coefficients [absolute value of ([a.sub.2])] and [absolute value of ([a.sub.3])].

2. Biunivalent Function Classes [M.sup.k.sub.[SIGMA]]([lambda],[PHI](z, t)) and [F.sup.k.sub.[SIGMA]]([beta],[PHI](z, t))

Motivated by recent works of Altinkaya and Yalcin  (also see ) and recent studies on biunivalent functions involving Salagean operator [11, 13], in this section, we introduce two new subclasses of Z associated with Chebyshev polynomials and obtain the initial Taylor coefficients [absolute value of ([a.sub.2])] and [absolute value of ([a.sub.3])] for the function classes by subordination.

Definition 1. For 0 [less than or equal to] [lambda] [less than or equal to] 1 and t [member of] (-1, 1) a function f [member of] [SIGMA] of form (1) is said to be in the class [M.sup.k.sub.[SIGMA]]([lambda], [PHI](z, t)) if the following subordination holds:

[mathematical expression not reproducible] (17)

where z, w [member of] [DELTA] and g is given by (4).

We note that by specializing the parameters [lambda] and suitably fixing the values for k in Definition 1, we introduce (had not been studied so far) the following new subclasses of [SIGMA] as listed below.

Remark 2. Supposing f(z) [member of] [SIGMA] and t [member of] (-1,1), then we denote

(1) [M.sup.k.sub.[SIGMA]](0, [PHI](z, t)) [equivalent to] [S.sup.k.sub.[SIGMA]]([PHI](z, t)),

(2) [M.sup.k.sub.[SIGMA]](1, [PHI](z, t)) [equivalent to] [K.sup.k.sub.[SIGMA]]([PHI](z, t)),

(3) [M.sup.0.sub.[SIGMA]](0, [PHI](z, t)) = [S.sup.*.sub.[SIGMA]]([PHI](z, t)),

(4) [M.sup.0.sub.[SIGMA]](1, [PHI](z, t)) = [K.sub.[SIGMA]]([PHI](z, t)).

Due to Frasin and Aouf  and Panigarhi and Murugusundaramoorthy  (also see [11, 13]) we define the following new subclass involving the Salagean operator .

Definition 3. For 0 [less than or equal to] [beta] [less than or equal to] 1 and t [member of] (-1,1) a function f [member of] [SIGMA] of form (1) is said to be in the class [F.sup.k.sub.[SIGMA]]([beta],[PHI](z,t)) if the following subordination holds:

(1 - [beta])[[D.sup.k]f(z)/z] + [beta]([D.sup.k]f(z))' [??] [PHI](z,t),

(1 - [beta])[[D.sup.k]g(w)/w] + [beta]([D.sup.k]g(w))' [??] [PHI](w,t), (18)

where z, w [member of] [DELTA], g = [f.sup.-1,] [D.sup.k]f(z) and [D.sup.k]g(w) are given by (4), (15), and (16), respectively.

In Definition 3, by specializing the parameters [beta] and suitably fixing the values for k (had not been studied so far) the following new subclasses of [SIGMA] are as listed below.

Remark 4. Supposing f(z) [member of] [SIGMA] and t [member of] (-1,1), then we denote

(1) [F.sup.k.sub.[SIGMA]](0,[PHI](z,t)) [equivalent to] [R.sup.k.sub.[SIGMA]]([PHI](z,t)),

(2) [F.sup.k.sub.[SIGMA]](1,[PHI](z,t)) [equivalent to] [H.sup.k.sub.[SIGMA]]([sigma](z,t)),

(3) [F.sup.0.sub.[SIGMA]]([beta],[PHI](z,t)) [equivalent to] [F.sub.[SIGMA]]([beta],[PHI](z,t)),

(4) [F.sup.0.sub.[SIGMA]](1,[PHI](z,t)) [equivalent to] [H.sup.0.sub.[SIGMA]]([sigma](z,t)).

In the following theorems we determine the initial Taylor coefficients [absolute value of ([a.sub.2])] and [absolute value of ([a.sub.2)] for the function classes f [member of] [M.sup.k.sub.[SIGMA]]([lambda], [PHI](z, t)) and f [member of] [F.sup.k.sub.[SIGMA]]([beta], [PHI](z, t)).

Theorem 5. Let f given by (1) be in the class [M.sup.k.sub.[SIGMA]]([lambda], [PHI](z, t)) and t [member of] (0,1). Then

[mathematical expression not reproducible] (19)

where 0 [less than or equal to] [lambda] [less than or equal to] 1 and t [not equal to] 1/[square root of 2].

Proof. Let f [member of] [M.sup.k.sub.[SIGMA]]([lambda], [PHI](z, t)) and g = [f.sup.-1]. Considering (17), we have

(1 - [lambda])[[D.sup.k+1]f(z)/[D.sup.k]f(z)] + [lambda] [[D.sup.k+2]f(z)/[D.sup.k+1]f(z)] = [PHI](z,t), (20)

(1 - [lambda])[[D.sup.k+1]g(w)/[D.sup.k]g(w)] + [lambda] [[D.sup.k+2]g(w)/[D.sup.k+1]g(w)] = [PHI](w,t), (21)

Define the functions u(z) and v(w) by

u(z) = [c.sub.1]z + [c.sub.2][z.sup.2] + ..., (22)

v(w) = [d.sub.1]w + [d.sub.2][w.sup.2] + ... (23)

which are analytic in [DELTA] with u(0) = 0 = v(0) and [absolute value of (u(z))] < 1, [absolute value of (v(w))] < 1, for all z [member of] [DELTA]. It is well known that

[mathematical expression not reproducible] (24)

and then

[absolute value of ([c.sub.j])] [less than or equal to] 1,

[absolute value of ([d.sub.j])] [less than or equal to] 1

[for all]j [member of] N. (25)

Using (22) and (23) in (20) and (21), respectively, we have

[mathematical expression not reproducible] (26)

In light of (1), (4), (10), (15), and (16) and from (26), we have

[mathematical expression not reproducible] (27)

This yields the following relations:

(1 + [lambda])[2.sup.k][a.sub.2] = [U.sub.1](t)[c.sub.1], (28)

-(1 + 3[lambda]) [2.sup.2k][a.sup.2.sub.2] + 2(1 + 2[lambda]) [3.sup.k][a.sub.3] = [U.sub.1](t)[c.sub.2] + [U.sub.2](t)[c.sup.2.sub.1], (29)

-(1 + [lambda])[2.sup.k][a.sub.2] = [U.sub.1](t)[d.sub.1], (30)

(4 (1 + 2[lambda]) [3.sup.k] - (1 + 3[lambda]) [2.sup.2k]) [a.sup.2.sub.2] - 2(1 + 2[lambda]) [3.sup.k][a.sub.3] = [U.sub.1](t) [d.sub.2] + [U.sub.2](t)[d.sup.2.sub.1]. (31)

From (28) and (30) it follows that

[c.sub.1] = -[d.sub.1], (32)

2 [(1 + [lambda]).sup.2] [2.sup.2k][a.sup.2.sub.2] = [U.sup.2.sub.1](t)([c.sup.2.sub.1] + [d.sup.2.sub.1]). (33)

Adding (29) to (31) and using (33), we obtain

[a.sup.2.sub.2]

= [U.sup.3.sub.1](t)([c.sub.2] + [d.sub.2])/2[{2 (1 + 2[lambda])[3.sup.k] - (1 + 3[lambda])[2.sup.2k]} [U.sup.2.sub.1](t) - [(1 + [lambda]).sup.2][2.sup.2k][U.sub.2](t)]. (34)

Applying (25) to the coefficients [c.sub.2] and [d.sub.2] and using (10) we have

[absolute value of ([a.sub.2])]

[less than or equal to] 2t[square root of 2t]/[square root of ([absolute value of ([2 (1 + 2[lambda])[3.sup.k] - ([[lambda].sup.2] + 5[lambda] + 2)[2.sup.2k]]4[t.sup.2] + [(1 + [lambda]).sup.2][2.sup.2k])])]. (35)

By subtracting (31) from (29) and using (32) and (33), we get

[a.sub.3] = [[U.sup.2.sub.1]([c.sup.2.sub.1] + [d.sup.2.sub.1])/2[(1 + [lambda]).sup.2] [2.sup.2k]] + [[U.sub.1]([c.sub.2] - [d.sub.2])/4(1 + 2[lambda])[3.sup.k]]. (36)

Using (10), once again applying (25) to the coefficients [c.sub.1], [c.sub.2], [d.sub.1], and [d.sub.2], we get

[absolute value of ([a.sub.3])] [less than or equal to] [4[t.sup.2]/[(1 + [lambda]).sup.2] [2.sup.2k]] + [t/(1 + 2[lambda])[3.sup.k]]. (37)

By taking [lambda] = 0 or [lambda] = 1 and 1 [member of] (0,1), one can easily state the estimates [absolute value of ([a.sub.2])] and [absolute value of ([a.sub.3])] for the function classes [M.sup.k.sub.[SIGMA]](0,[PHI](z,t)) = [S.sup.k.sub.[SIGMA]]([PHI](z,t)) and [M.sup.k.sub.[SIGMA]](1,[PHI](z,t)) = [K.sup.k.sub.[SIGMA]]([PHI](z, t)), respectively.

Remark 6. Let f given by (1) be in the class [S.sup.k.sub.[SIGMA]]([PHI](z, t)). Then

[mathematical expression not reproducible] (38)

Remark 7. Let f given by (1) be in the class [K.sup.k.sub.[SIGMA]]([PHI](z, t)). Then

[absolute value of ([a.sub.2])] [less than or equal to] [2t[square root of (2t)]/[square root of ([absolute value of ([[3.sup.k+1] - [2.sup.2(k+1)]] 8[t.sup.2] + [2.sup.2(k+2)])])], (39)

[absolute value of ([a.sub.3])] [less than or equal to] [[t.sup.2]/[2.sup.2k]] + [t/[3.sup.k+1]]. (40)

For k = 0, Theorem 5 yields the following corollary.

Corollary 8. Let f given by (1) be in the class [M.sup.0.sub.[SIGMA]]([lambda], [PHI](z, t)). Then

[mathematical expression not reproducible] (41)

where 0 [less than or equal to] [lambda] [less than or equal to] 1 and t [not equal to] 1/[square root of 2].

By taking k = 0 in the above remarks we get the estimates [absolute value of ([a.sub.2])] and [absolute value of ([a.sub.3])] for the function classes [S.sup.*.sub.[SIGMA]]([PHI](z, t)) and [K.sub.[SIGMA]]([PHI](z,t).

Remark 9. Let f given by (1) be in the class [S.sup.k.sub.[SIGMA]]([PHI](z, t)). Then

[absolute value of ([a.sub.2])] [less than or equal to] 2t/[square root of 2t],

[absolute value of ([a.sub.3])] [less than or equal to] 4[t.sup.2] + t. (42)

Remark 10. Let f given by (1) be in the class [K.sup.k.sub.[SIGMA]]([PHI](z,t)). Then, for t [not equal to] 1/[square root of 2],

[absolute value of ([a.sub.2])] [less than or equal to] 2t[square root of 2t]/[square root of ([absolute value of (4 - 8[t.sup.2])])],

[absolute value of ([a.sub.3])] [less than or equal to] [t.sup.2] + t/3. (43)

Theorem 11. Let f given by (1) be in the class [F.sup.k.sub.[SIGMA]]([beta], [PHI](z,t)) and 1 [member of] (0,1). Then

[absolute value of ([a.sub.2])]

[less than or equal to] 2t[square root of 2t]/[square root of ([absolute value of ([(1 + 2[beta]) [3.sup.k] - [(1 + [beta]).sup.2] [2.sup.2k]]4[t.sup.2] + [(1 + [beta]).sup.2] [2.sup.2k])])] (44)

[absolute value of ([a.sub.3])] [less than or equal to] [4[t.sup.2]/[(1 + [beta]).sup.2][2.sup.2k]] + [2t/(1 + 2[beta])[3.sup.k]]. (45)

Proof. Proceeding as in the proof of Theorem 5 we can arrive at the following relations:

(1 + [beta]) [2.sup.k][a.sub.2] = [U.sub.1](t)[c.sub.1], (46)

(1 + 2[beta]) [3.sup.k][a.sub.3] = [U.sub.1](t)[c.sub.2] + [U.sub.2](t) [c.sup.2.sub.1], (47)

-(1 + [beta])[2.sup.k][a.sub.2] = [U.sub.1](t)[d.sub.2], (48)

2(1 + 2[beta]) [3.sup.k][a.sup.2.sub.2] - (1 + 2[beta]) [3.sup.k][a.sub.3] = [U.sub.1](t)[d.sub.2] + [U.sub.2](t)[d.sup.2.sub.1]. (49)

From (46) and (48) it follows that

[c.sub.1] = -[d.sub.1], (50)

2[(1 + [beta]).sup.2] [2.sup.2k][a.sup.2.sub.2] = [U.sup.2.sub.1](t)([c.sup.2.sub.1] + [d.sup.2.sub.1]). (51)

From (47), (49), and (51), we obtain

[a.sup.2.sub.2] = [U.sup.3.sub.1](t)([c.sub.2] + [d.sub.2])/2[(1 + 2[beta])[3.sup.k][U.sup.2.sub.1](t) - [(1 + [beta]).sup.2] [2.sup.2k][U.sub.2](t)]. (52)

Using (10) and (25) for the coefficients [c.sub.2] and [d.sub.2], we immediately get the desired estimate on [absolute value of ([a.sub.2])] as asserted in (44).

By subtracting (49) from (47) and using (50) and (51), we get

[a.sub.3] = [[U.sup.2.sub.1](t)([c.sup.2.sub.1] + [d.sup.2.sub.1])/2[(1 + [beta]).sup.2][2.sup.2k]] + [U.sub.1](t)([c.sub.2] - [d.sub.2])/2(1 + 2[beta])[3.sup.k]. (53)

Again using (10) and (25) for the coefficients [c.sub.1], [c.sub.2], [d.sub.1], and [d.sub.2], we get the desired estimate on [absolute value of ([a.sub.3])] as asserted in (45).

Remark 12. Let f given by (1) be in the class [R.sup.k.sub.[SIGMA]]([PHI](z, t)). Then

[absolute value of ([a.sub.2])] [less than or equal to] 2t[square root of 2t]/[square root of ([absolute value of (([3.sup.k] - [2.sup.2k])4[t.sup.2] + [2.sup.2k])])],

[absolute value of ([a.sub.3])] [less than or equal to] [4[t.sup.2]/[2.sup.2k]] + [2t/[3.sup.k]]. (54)

Remark 13. Let f given by (1) be in the class [H.sup.k.sub.[SIGMA]]([PHI](z, t)). Then

[absolute value of ([a.sub.2])] [less than or equal to] 2t[square root of 2t]/[square root of ([absolute value of (([3.sup.k+1] - [2.sup.2k+1])4[t.sup.2] + [2.sup.2(k+1)])])],

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

By taking k = 0 we deduce the following results.

Remark 14. Let f given by (1) be in the class [F.sub.[SIGMA]]([beta], [PHI](z, t)). Then

[absolute value of ([a.sub.2])] [less than or equal to] 2t[square root of 2t]/[square root of ([absolute value of ([(1 + [beta]).sup.2] - 4[t.sup.2][[beta].sup.2])])]

[absolute value of ([a.sub.3])] [less than or equal to] [4[t.sup.2]/[(1 + [beta]).sup.2]] + [2t/(l + [beta])]. (56)

Remark 15. Let f given by (1) be in the class [F.sup.0.sub.[SIGMA]](1, [PHI](z, t)) [equivalent to] [H.sub.[SIGMA]]([PHI](z,t))* Then

[absolute value of ([a.sub.2])] [less than or equal to] t[square root of 2t]/[square root of ([absolute value of (1 - [t.sup.2])])],

[absolute value of ([a.sub.2])] [less than or equal to] [t.sup.2] + [2t/3]. (57)

Remark 16. Let f given by (1) be in the class [F.sup.0.sub.[SIGMA]](0, [PHI](z, t)) [equivalent to] [R.sub.[SIGMA]]([PHI](z, t)). Then

[absolute value of ([a.sub.2])] [less than or equal to] 2t[square root of 2t],

[absolute value of ([a.sub.3])] [less than or equal to] 4[t.sup.2] + 2t. (58)

3. Fekete-Szego Inequality for the Function Classes [M.sup.k.sub.[SIGMA]]([lambda], [PHI](z, t)) and [F.sup.k.sub.[SIGMA]]([lambda], [PHI](z, t))

Due to Zaprawa , in this section we obtain the Fekete-Szego inequality for the function classes [M.sup.k.sub.[SIGMA]]([lambda], 0(z, t)) and [F.sup.k.sub.[SIGMA]]([beta], [PHI](z,t)).

Theorem 17. Let f given by (1) be in the class [M.sup.k.sub.[SIGMA]]([lambda], [PHI](z, t)) and [mu] [member of] R. Then one has

[mathematical expression not reproducible] (59)

Proof. From (29) and (31)

[mathematical expression not reproducible] (60)

where

h([mu])

(1 - [mu])[U.sup.2.sub.1](t)/2[(2(1 + 2[lambda])[3.sup.k] - (1 + 3[lambda])[2.sup.2k])[U.sup.2.sub.1](t) - [(1 + [lambda]).sup.2][2.sup.k][U.sub.2](t)]. (61)

Then, in view of (10), we conclude that

[mathematical expression not reproducible] (62)

Taking [mu] = 1, we have the following corollary.

Corollary 18. If f [member of] [M.sup.k.sub.[SIGMA]]([lambda], [PHI](z, t)), then

[absolute value of ([a.sub.3] - [a.sup.2.sub.2])] [less than or equal to] t/(1 + 2[lambda])[3.sup.k]. (63)

Corollary 19. Let f given by (1) be in the class [S.sup.k.sub.[SIGMA]]([PHI](z, t)) and [mu] [member of] R. Then one has

[mathematical expression not reproducible] (64)

Particularly, for [mu] = 1 if f [member of] [S.sup.*.sub.[SIGMA]]([PHI](z,t)) one obtains

[absolute value of ([a.sub.3] - [a.sup.2.sub.2])] [less than or equal to] t. (65)

Corollary 20. Let f given by (1) be in the class [K.sup.k.sub.[SIGMA]]([PHI](z, t)) and [mu] [member of] R. Then one has

[mathematical expression not reproducible] (66)

Particularly, for [mu] = 1 if f [member of] [K.sup.0.sub.[SIGMA]]([PHI](z,t)) one obtains

[absolute value of ([a.sub.3] - [a.sub.2])] [less than or equal to] [t/3]. (67)

Theorem 21. Let f given by (1) be in the class [F.sup.k.sub.[SIGMA]]([beta], [PHI](z, t)) and [mu] [member of] R. Then one has

[mathematical expression not reproducible] (68)

Proof. From (29) and (31)

[mathematical expression not reproducible] (69)

where

h([mu]) = (1 - [mu])[U.sup.2.sub.1](t)/2[(1 + 2[beta])[3.sup.k][U.sup.2.sub.1](t) - [(1 + [beta]).sup.2][2.sup.2k] [U.sub.2](t)]. (70)

Then, in view of (10), we conclude that

[mathematical expression not reproducible] (71)

Taking [mu] = 1, we have the following corollary.

Corollary 22. If f [member of] [F.sup.k.sub.[SIGMA]]([beta], [PHI](z, t)), then

[absolute value of ([a.sub.3] - [a.sup.2.sub.2])] [less than or equal to] [2t/(1 + 2[beta][3.sup.k])]. (72)

Corollary 23. Let f given by (1) be in the class [R.sup.k.sub.[SIGMA]]([PHI](z,t)) and [mu] [member of] R. Then one has

[mathematical expression not reproducible] (73)

Particularly, for [mu] = 1 if f [member of] [R.sup.0.sub.[SIGMA]]([PHI](z, t)) one obtains

[absolute value of ([a.sub.3] - [a.sup.2.sub.2]) [less than or equal to] 2t. (74)

Corollary 24. Let f given by (1) be in the class [H.sup.k.sub.[SIGMA]]([PHI](z, t)) and [mu] [member of] R. Then one has

[mathematical expression not reproducible] (75)

Particularly, for [mu] = 1 if f [member of] [H.sup.0.sub.[SIGMA]]([PHI](z, t)) one obtains

[absolute value of ([a.sub.3] - [a.sup.2.sub.2] [less than or equal to] [2t/3]. (76)

https://doi.org/10.1155/2017/4150210

Conflicts of Interest

The authors declare that they have no conflicts of interest regarding the publication of this paper.

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Hatun Ozlem Guney, (1) G. Murugusundaramoorthy, (2) and K. Vijaya (2)

(1) Faculty of Science, Department of Mathematics, Dicle University, 21280 Diyarbakir, Turkey

(2) School of Advanced Sciences, VIT University, Vellore 632014, India

Correspondence should be addressed to Hatun (Ozlem Guney; ozlemg@dicle.edu.tr

Received 28 August 2017; Accepted 26 October 2017; Published 26 November 2017

Academic Editor: Yan Xu
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