# Application of Quasi-Subordination for Generalized Sakaguchi Type Functions.

1. Introduction and Definitions

Let [DELTA] be the unit disk

{z : z [member of] C, [absolute value of z] < 1}, (1)

and let A be the class of functions analytic in [DELTA], satisfying the conditions

f(0) = 0,

f'(0) = 1. (2)

Then each f [member of] A has the Taylor expansion

f(z) = z + [[infinity].summation over (k=2)][a.sub.k][z.sup.k]. (3)

Moreover, by S we shall represent the class of all functions in A which are univalent in [DELTA]. Let h(z) be an analytic function in [DELTA] and [absolute value of (h(z))] [less than or equal to] 1, such that

h(z) = [h.sub.0] + [h.sub.1]z + [h.sub.2][z.sup.2] + ..., (4)

where all coefficients are real. Also, let [phi] be an analytic and univalent function with positive real part in [DELTA] with [phi](0) = 1, [phi]'(0) > 0, and [phi] maps the unit disc [DELTA] onto a region starlike with respect to 1 and symmetric with respect to the real axis. Taylor's series expansion of such function is of the form

[phi](z) = 1 + [B.sub.1]z + [B.sub.2][z.sup.2] + ..., (5)

where all coefficients are real and [B.sub.1] > 0. Let P be the class of functions consisting of form (5).

If the functions f and g are analytic in [DELTA], then f is said to be subordinate to g, written as

f(z) [??] g(z), (z [member of] [DELTA]) (6)

if there exists a Schwarz function w(z), analytic in [DELTA], with

w(0) = 0,

[absolute value of (w(z))] < 1 (z [member of] [DELTA]) (7)

such that

f(z) = g(w(z)) (z [member of] A). (8)

In the year 1970, Robertson [1] introduced the concept of quasi-subordination. For two analytic functions f and g, the function f is said to be quasi-subordinate to g in [DELTA] and written as

f(z) [[??].sub.q] g(z) (z [member of] A) (9)

if there exists an analytic function [absolute value of h(z))] [less than or equal to] 1 such that f(z)fh(z) analytic in [DELTA] and

f(z)/h(z) [??] g(z) (z [member of] [DELTA]); (10)

that is, there exists a Schwarz function w(z) such that f(z) = h(z)g(w(z)). Observe that if h(z) = 1, then f(z) = g(w(z)) so that f(z) [??] g(z) in A. Also notice that if w(z) = z, then f(z) = h(z)g(z) and it is said that it is majorized by g and written by f(z) [much less than] g(z) in [DELTA]. Hence it is obvious that quasi-subordination is a generalization of subordination as well as majorization (see, e.g., [1-3] for works related to quasi-subordination).

In [4], Sakaguchi introduced the class [S.sup.*.sub.S] of starlike functions with respect to symmetric points in A, consisting of functions f [member of] A that satisfy the condition R(zf'(z)/(f(z) - f(-z))) > 0, z [member of] [DELTA]. Similarly, in [5], Wang et al. introduced the class [C.sub.S] of convex functions with respect to symmetric points in [DELTA], consisting of functions f [member of] [DELTA] that satisfy the condition R((zf'(z))'/(f'(z) + f'(-z))) > 0, z [member of] [DELTA]. For different parametric values, we get the classes studied in the literature by Frasin [6], Goyal et al. [7], and Owa et al. [8].

The Fekete-Szego functional [absolute value of ([a.sub.3] - [mu][a.sup.2.sub.2])] for normalized univalent functions of the form given by (3) is well known for its rich history in Geometric Function Theory. Its origin was in the disproof by Fekete and Szego [9] of the 1933 conjecture of Littlewood and Paley that the coefficients of odd univalent functions are bounded by unity (see, for details, [9]). The Fekete-Szego functional has [absolute value of ([a.sub.3] - [mu][a.sup.2.sub.2])] since it received great attention, particularly in connection with many subclasses of the class of normalized analytic and univalent functions (see, e.g., [10-15]).

The object of the present paper is to introduce a new class of univalent functions applying the Ruscheweyh derivative, where Ruscheweyh [16] observed that

[D.sup.n]f(z) = z[[[z.sup.n-1]f(z)].sup.(n)]/n! (11)

for n [member of] [N.sub.0] = N [union] {0}, where N = {1, 2, ...}. This symbol [D.sup.n]f(z), n [member of] [N.sub.0], is called by Al-Amiri [17] the nth order Ruscheweyh derivative of f(z). We note that [D.sup.0] f(z) = f(z), [D.sup.1]f(z) = zf'(z), and

[mathematical expression not reproducible] (12)

2. Preliminary Results

The study of special functions plays an important role in Geometric Function Theory in Complex Analysis and its related fields. Special functions can be categorized into three, namely, Ramp function, threshold function, and sigmoid function. The popular type among all is the sigmoid function because of its gradient descendent learning algorithm. It can be evaluated in different ways, most especially by truncated series expansion. The sigmoid function of the form

[kappa](z) = [1/1 + [e.sup.-z]] (13)

is useful because it is differentiable. The Sigmoid function has very important properties, including the following (see [18]):

(i) It outputs real numbers between 0 and 1.

(ii) It maps a very large input domain to a small range of outputs.

(iii) It never loses information because it is a one-to-one function.

(iv) It increases monotonically.

Lemma 1 (see [19]). Let the Schwarz function w(z) be given by

w(z) = [w.sub.1]z + [w.sub.1][z.sup.2] + ..., (z [member of] [DELTA]); (14)

then

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

[absolute value of ([w.sub.2] - I[w.sup.2.sub.1])] [less than or equal to] 1 + ([absolute value of I] - 1) [[absolute value of ([w.sub.1])].sup.2] [less than or equal to] max {1, [absolute value of I]}, (15)

where I [member of] C.

Lemma 2 (see [18]). Let k be a sigmoid function and

G(z) = 2[kappa](z) = 1 + [[infinity].summation over (m=1)][[(-1).sup.m]/[2.sup.m]]([[infinity].summation over (n=1)][[(-1).sup.n]/n!][z.sup.n]), (16)

and then G(z) [member of] P, [absolute value of z] < 1, where G(z) is a modified sigmoid function.

Lemma 3 (see [18]). Let

[G.sub.n,m](z) = 1 + [[infinity].summation over (m=1)][[(-1).sup.m]/[2.sup.m]][([[infinity].summation over (n=1)][[(-1).sup.n]/n!][z.sup.n]).sup.m], (17)

and then [absolute value of ([G.sub.n,m](z))] < 2.

3. Main Result and Its Consequences

Definition 4. A function f [member of] [DELTA] is said to be in the class [S.sup.[lambda].sub.q] G, s, t), if the following quasi-subordination holds:

[([D.sup.n]f(z))'[((s - t)z/[D.sup.n]f(sz) - [D.sup.n]f(tz)).sup.[lambda]] - 1]

[[??].sub.q] (G(z) - 1), z [member of] [DELTA], (18)

where s, t [member of] C with s [not equal to] t, [absolute value of t] [less than or equal to] 1 and [lambda] [greater than or equal to] 0.

From the definition, it follows that f [member of] [S.sup.[lambda].sub.q](G, s, t) if and only if there exists an analytic function h(z) with [absolute value of (h(z))] [less than or equal to] 1, such that

[([D.sup.n]f(z))'((s - t)z/([D.sup.n]f(sz) - [D.sup.n]f(tz)).sup.[lambda]] - 1/h(z) [??] (G(z) - 1). (19)

If, in the subordination condition (19), h(z) = 1, then the class [S.sup.[lambda].sub.q](G, s, t) is denoted by [S.sup.[lambda]](G, s, t) and the functions therein satisfy the condition that

([D.sup.n]f(z))' [((s - t)z/[D.sup.n]f(sz) - [D.sup.n]f(tz)).sup.[lambda]] [??] G(z), z [member of] [DELTA]. (20)

Theorem 5. Let f of the form (3) be in the class [S.sup.[lambda].sub.q](G,s,t).

Then

[absolute value of ([a.sub.2])] [less than or equal to] [1/2(n + 1)[absolute value of (2 - [lambda](s + t))] (21)

and for some [eta] [member of] C

[mathematical expression not reproducible] (22)

and the result is sharp.

Proof. Let f [member of] [S.sup.[lambda].sub.q](G,s,t). In view of Definition 4, we can write

([D.sup.n]f(z))' [((s - t)z/[D.sup.n]f(sz) - [D.sup.n]f(tz)).sup.[lambda]] - 1 [??] h(z)(G(w(z)) - 1), (23)

where the function G(z) is a modified sigmoid function given by

G(z) = 1 + [1/2]z - [1/24][z.sup.3] + [1/240][z.sup.5] - [1/64][z.sup.6] + [779/20160][z.sup.7] - .... (24)

Combining (4), (14), and (24), we obtain

h(z)(G(w(z)) - 1) = [1/2][h.sub.o][w.sub.1]z + [1/2]([h.sub.o][w.sub.2] + [h.sub.1][w.sub.1])[z.sup.2] + .... (25)

In the light of (23) and (25), we get

(n + 1) [2 - [lambda] (s + t)] [a.sub.2] = [1/2][h.sub.o][w.sub.1], (26)

[mathematical expression not reproducible] (27)

Now, (26) gives

[a.sub.2] = [h.sub.o][w.sub.1]/2(n + 1)[2 - [lambda](s + t)]. (28)

From (27), it follows that

[mathematical expression not reproducible] (29)

For some [eta] [member of] C, we obtain from (28) and (29)

[mathematical expression not reproducible] (30)

Since h(z) given by (4) is analytic and bounded in [DELTA], therefore, on using [20] (p 172), we have for some y ([absolute value of y] [less than or equal to] 1)

[absolute value of ([h.sub.o])] [less than or equal to] 1, [h.sub.1] = (1 - [h.sup.2.sub.o]) y. (31)

On putting the value of [h.sub.1] from (31) into (30), we get

[mathematical expression not reproducible] (32)

If [h.sub.o] = 0 in (32), we obtain

[absolute value of ([a.sub.3] - [eta][a.sup.2.sub.2])] [less than or equal to] 1/(n + 1) (n + 2) [absolute value of (3 - [lambda] ([s.sup.2] + st + [t.sup.2]))]. (33)

If [h.sub.o] [not equal to] 0 in (32), let

[mathematical expression not reproducible] (34)

which is a polynomial in [h.sub.o] and hence analytic in [absolute value of ([h.sub.o])] [less than or equal to] 1, and maximum [absolute value of (T([h.sub.o]))] is attained at [h.sub.o] = [e.sup.i[theta]], (0 [less than or equal to] [theta] < 2[pi]). We find that

[mathematical expression not reproducible] (35)

which on using Lemma 1 shows that

[mathematical expression not reproducible] (36)

For the case when s = 1, one has the following.

Corollary 6. Let f of form (3) be in the class [S.sup.[lambda].sub.q](G, 1, t). Then

[absolute value of ([a.sub.2])] [less than or equal to] [1/2(n + 1)[absolute value of (2 - [lambda](1 + t))]] (37)

and for some [eta] [member of] C

[mathematical expression not reproducible] (38)

and the result is sharp.

Putting t = -1 in Corollary 6, we obtain the following corollary.

Corollary 7. Let f of form (3) be in the class [S.sup.[lambda].sub.q](G, 1, -1). Then

[absolute value of ([a.sub.2])] [less than or equal to] [1/4(n + 1)] (39)

and for some [eta] [member of] C

[mathematical expression not reproducible] (40)

and the result is sharp.

Setting [lambda] = 0 in Corollary 7, we have the following.

Corollary 8. Let f of form (3) be in the class [S.sub.q](G, 1, -1). Then

[absolute value of ([a.sub.2])] [less than or equal to] [1/4(n + 1)] (41)

and for some [eta] [member of] C

[absolute value of ([a.sub.3] - [eta][a.sup.2.sub.2])] [less than or equal to] [1/3(n + 1)(n + 2)] max{1, [absolute value of (3[eta](n + 2)/16(n + 1)]} (42)

and the result is sharp.

Setting [lambda] = 1 in Corollary 7, we have the following.

Corollary 9. Let f of form (3) be in the class [S.sub.q](G, 1, -1). Then

[absolute value of ([a.sub.2])] [less than or equal to] [1/4(n + 1)] (43)

and for some [eta] [member of] C

[absolute value of ([a.sub.3] - [eta][a.sup.2.sub.2])] [less than or equal to] [1/2(n + 1)(n + 2)] max{1, [absolute value of ([eta](n + 2)/8(n + 1)]} (44)

and the result is sharp.

https://doi.org/10.1155/2017/3780675

Conflicts of Interest

The author declares that they have no conflicts of interest.

Acknowledgments

This research is supported by the Scientific and Technological Research Council of Turkey (TUBITAK 2214A).

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Fahsene Altinkaya

Department of Mathematics, Faculty of Arts and Science, Uludag University, 16059 Bursa, Turkey

Correspondence should be addressed to Sahsene Altinkaya; sahsenealtinkaya@gmail.com

Received 19 August 2017; Accepted 8 October 2017; Published 31 October 2017