# Geometric Properties of Cesaro Averaging Operators.

1. Introduction

Inequalities involving trigonometric sums arise naturally in various problems of pure and applied mathematics. Inequalities that assure nonnegativity or boundedness of partial sums of trigonometric series are of particular interest and applications in various fields. For example, the positivity of trigonometric polynomials are studied in geometric function theory by Gluchoff and Hartmann  and Ruscheweyh and Salinas . For a detailed application in signal processing, we refer to the monograph of Dumitrescu . For other applications in this direction, we refer to Dimitrov and Merlo , Fernandez-Duran , and Gasper . The positive trigonometric polynomials played an important role in the proof of Bieberbach conjecture; see . For the applications of positive trigonometric polynomials in Fourier series, approximation theory, function theory, and number theory, we refer to the work of Dimitrov  and references therein. For the study of extremal problems, we refer to the dissertation of Revesz  wherein several applications are outlined.

The problem of finding the behaviour of the coefficients to validate the positivity of trigonometric sum has been dealt by many researchers. Among them, the contributions of Vietoris  followed by Koumandos  are of interest to the present investigation. Precisely, Vietoris  gave sufficient conditions on the coefficient of a general class of sine and cosine sums that ensure their positivity in (0, [pi]). For further details in this direction one can refer to [11-13] and the references therein. An account of recent results available in this direction is given in  and one of the main results in  is as follows.

Theorem 1 (see ). Suppose that [alpha] [greater than or equal to] 0, [beta] [greater than or equal to] 0, and [lambda], [mu] [greater than or equal to] 0 such that [lambda] + [mu] [greater than or equal to] 1 then for [b.sub.0] = 2, [b.sub.1] = 1, and [b.sub.k] = 1/[(k + [alpha]).sup.[lambda]][(k + [beta]).sup.[mu]], k [greater than or equal to] 2, we have

[b.sub.0]/2 + [n.summation over (k=1)][b.sub.k]cosk[theta] > 0,

[n.summation over (k=1)][b.sub.k]sink[theta] > 0, (1)

for 0 < [theta] < [pi] and n [member of] N.

Using summation by parts, the following corollary of Theorem 1 can be obtained.

Corollary 2. For [alpha] [greater than or equal to] 0, [beta] [greater than or equal to] 0 and [lambda] [greater than or equal to] 0, [lambda] [greater than or equal to] 0 such that [lambda] + [mu] [greater than or equal to] 1. If there exists a sequence {[a.sub.k]} of positive real numbers, such that

[mathematical expression not reproducible] (2)

then, for n [member of] N, the following inequalities hold:

[a.sub.0]/2 + [n.summation over (k=1)][a.sub.k]cosk[theta] > 0,

[n.summation over (k=1)][a.sub.k]sink[theta] > 0, (3)

where 0 < [theta] < [pi].

The main purpose of this note is to use Corollary 2 to find certain geometric properties of analytic functions, in particular univalent functions. Let [A.sub.0] be the subclass of the class of analytic functions f [member of] A with normalized conditions f(0) = 0, f'(0) = 1 in the unit disc D = {z [member of] D, [absolute value of z] < 1}. The subclasses of [A.sub.0] consisting of univalent function are denoted by S. Several subclasses of univalent functions play a prominent role in the theory of univalent functions. For 0 [less than or equal to] [gamma] < 1, let [S.sup.*]([gamma]) be the family of functions f starlike of order [gamma]; that is, if f [member of] [A.sub.0] satisfies the analytic characterization,

f [member of] [S.sup.*]([gamma]) [??] Re (zf'(z)/f(z)) > [gamma], for z [member of] D. (4)

For 0 [less than or equal to] [gamma] < 1, let C([gamma]) be the family of functions f convex of order [gamma]; that is, if f [member of] [A.sub.0] satisfies the analytic characterization,

f [member of] C([gamma]) [??]

Re(1 + [zf"(z)/f'(z)]) > [gamma], for z [member of] D. (5)

These two classes are related by the Alexander transform, f [member of] C([gamma]) [??] zf' [member of] [S.sup.*]([gamma]). The usual classes of starlike functions (with respect to origin) and convex functions are denoted, respectively, by [S.sup.*](0) [equivalent to] [S.sup.*] and C(0) [equivalent to] C. An analytic function f is said to be close-to-convex of order [gamma], (0 [less than or equal to] [gamma] < 1) with respect to a fixed starlike function g if and only if it satisfies the analytic characterization:

Re [e.sup.i[eta]]([zf'(z)/g(z)] - [mu]) > 0,

z [member of] D, [eta] [member of] (-[[pi]/2], [pi], 2), g [member of] [S.sup.*]. (6)

The family of all close-to-convex function of order [mu] with respect to g [member of] [S.sup.*] is denoted by [K.sub.g]([mu]). Further, for 0 [less than or equal to] [mu] < 1, each f [member of] [K.sub.g]([mu]) is also univalent in D. The proper inclusion between these classes is given by

[mathematical expression not reproducible] (7)

Another important subclass is the class of typically real functions. A function f [member of] [A.sub.0] is typically real if Im(z)Im(f(z)) [greater than or equal to] 0 where z [member of] D. Its class is denoted by T. For several interesting geometric properties of these classes, one can refer to the standard monographs [14-16] on univalent functions.

Remark 3. The functions

z, [z/1 [+ or -] z], [z/1 [+ or -] [z.sup.2]], [z/[(1 [+ or -] z).sup.2], [z/(1 [+ or -] z [+ or -] [z.sup.2])] (8)

are the only nine starlike univalent functions having integer coefficients in D. It will be interesting to find f to be close-to-convex when the corresponding starlike function g takes one of the above forms.

If we take [eta] = 0 and g(z) = z/[(1 - z).sup.2] then Re([(1 - z).sup.2]f'(z)) > 0 which implies zf'(z) is typically real function. A function f [member of] [A.sub.0] is said to be typically real if Im f(z)Im(z) > 0 whenever Im(z) [not equal to] 0, z [member of] D. The function [k.sub.[gamma]](z) := z/[(1 - z).sup.2-2[gamma]] is the extremal function for the class of starlike function of order [gamma]. Note that [k.sub.0](z) is the well-known Koebe function and the function [k.sub.1/2](z) = z/(1 - z) is the extremal function for the class C. A function f(z) is said to be prestarlike of order [gamma], 0 [less than or equal to] [gamma] < 1, if [k.sub.[gamma]](z) * f(z) = z/[(1 - z).sup.2] * f(z) [member of] [S.sup.*]([gamma]) where "*" is the convolution operator or Hadamard product. This class was introduced by Ruscheweyh . For more details of this class see . Here the Hadamard product or convolution is defined as follows: let f(z) = [[summation].sup.[infinity].sub.k=0] [a.sub.k][z.sup.k] and g(z) = [[summation].sup.[infinity].sub.k=0] [b.sub.k][z.sup.k], z [member of] D. Then,

(f * g)(z) = [[summation].sup.[infinity].sub.k=0][a.sub.k][b.sub.k][z.sup.k], z [member of] D. (9)

Among all applications of positivity of trigonometric polynomials, the geometric properties of the subclasses of analytic functions are considered in this note. In this direction, Ruscheweyh  obtained some coefficient conditions for the class of starlike functions using the classical result of Vietoris . So it would be interesting to find the geometric properties of function f(z) in which Corollary 2 plays a vital role.

2. Geometric Properties of an Analytic Function

In this section, we provide conditions on the Taylor coefficients of an analytic function f to guarantee the admissibility of f in subclasses of S, using Corollary 2. The next lemma which is the generalization of [19, Lemma 2] is the crucial ingredient in the proof of the following theorem.

Lemma 4 (see [20, Theorem 3.1]). Let 0 [less than or equal to] [gamma] < 1 and f [member of] A be such that f'(z) and f'(z) - [gamma](f(z)/z) are typically real in D. Further if Re f'(z) > 0 and Re(f'(z) - [gamma](f(z)/z)) > 0, then f [member of] [S.sup.*]([gamma]).

Theorem 5. Let [alpha] [greater than or equal to] 0, [beta] [greater than or equal to] 0, [lambda] [greater than or equal to] 0, [mu] [greater than or equal to] 0 such that [lambda] + [mu] [greater than or equal to] 1; let [{[a.sub.k]}.sup.[infinity].sub.k=1] be any sequence of positive real numbers such that [a.sub.1] = 1. Let {[a.sub.k]} satisfy the following conditions:

(1) (2 - [gamma])[a.sub.2] [less than or equal to] (1 - [gamma])[a.sub.1].

(2) (3 - [gamma])[a.sub.3] [less than or equal to] [(1/(2 + [alpha]).sup.[lambda]] [(2 + [beta]).sup.[mu]]) (2 - [gamma]y)[a.sub.2].

(3) (k + 2 - [gamma])[a.sub.k+2] [less than or equal to] [(1 + 1/(k + [alpha])).sup.-[lambda] [(1 + 1/(k + [beta])).sup.-[mu]](k + 1 - [gamma])[a.sub.k+1], [for all]k [greater than or equal to] 2.

Then, for 0 [less than or equal to] [gamma] < 1, [f.sub.n](z) = z + [[summation].sup.n.sub.k=1][a.sub.k][z.sup.k] and f(z) = [lim.sub.n[right arrow][infinity]][f.sub.n](z) = z + [[summation].sup.[infinity].sub.k=2][a.sub.k][z.sup.k] are starlike of order [gamma].

Proof. Let [f.sub.n](z) = z + [[summation].sup.n.sub.k=2][a.sub.k][z.sup.k], z [member of] D be the partial sum of f. Then [f'.sub.n](z) = 1 + [[summation].sup.n-1.sub.k=1](k + 1)[a.sub.k+1][z.sup.k].

Define

[g.sub.n](z) := [f'.sub.n](z) - [gamma] [[f.sub.n](z)/z] = [[b.sub.0]/2] + [n-1.summation over (k=1)][b.sub.k][z.sup.k], z [member of] D, (10)

where [b.sub.0] = 2(1 - [gamma]) and [b.sub.k] = (k + 1 - [gamma])[a.sub.k+1], [for all]k [greater than or equal to] 1. Consider,

[mathematical expression not reproducible] (11)

Now, for k [greater than or equal to] 2,

[mathematical expression not reproducible] (12)

So by the given hypothesis, {[b.sub.k]} satisfy the conditions of Corollary 2 which implies Re [g.sub.n](z) > 0 and Im [g.sub.n](z) > 0 if Im(z) > 0. By reflection principle Im [g.sub.n](z) < 0 if Im(z) < 0. So [g.sub.n](z) is typically real function. In order to prove the theorem it is remaining to show that Re [f'.sub.n](z) > 0 and [f'.sub.n](z) is typically real. In this case [b.sub.k] = (k + 1)[a.sub.k+1] and [b.sub.0] = 2. So such [b.sub.k] also satisfy given hypothesis because (k + 1 - [gamma])/(k + 2 - [gamma]) < (k + 1)/(k + 2), for all k [greater than or equal to] 0. So Re [f'.sub.n](z) > 0 and again using reflection principle we get that [f'.sub.n](z) is typically real in D.

Applying Lemma 4, we get that [f.sub.n](z) [member of] [S.sup.*]([gamma]). Since [lim.sub.n[right arrow][infinity]][f.sub.n](z) = f(z) and the family of starlike functions is normal [21,p. 217],we get f(z) = [lim.sub.n[right arrow][infinity]] fn(z) is also starlike of order [gamma].

Remark 6. If [gamma] = 0 in Theorem 5, then we get Re([f'.sub.n](z)) > 0 which implies [f.sub.n](z) is close-to-convex with respect to z and [f'.sub.n](z) is typically real also and with

Re (1 - z)[f'.sub.n](z) = Re (1 - z) Re [f'.sub.n](z) + Im (z) Im [f'.sub.n](z) > 0 (13)

this yields [f.sub.n](z) that is close-to-convex with respect to starlike function z/(1 - z).

Example 7. Consider the sequence {[a.sub.k]} as [a.sub.1] = 1, [a.sub.2] = 1/2, and [a.sub.k] = 1/[k.sup.2] for k [greater than or equal to] 3; then by Theorem 5, the function

f(z) = z + [[z.sup.2]/2] + [[summation].sup.[infinity].sub.k=3] [[z.sup.k]/[k.sup.2]], z [member of] D (14)

is starlike univalent. But [22, Theorem 2.1] fails to include this function. Hence Theorem 5 is better than [22, Theorem 2.1] in the sense that it is likely to include more cases.

By proving that z[f'.sub.n](z) is typically real function in the similar fashion, we obtain the next result.

Theorem 8. Let [alpha] [greater than or equal to] 0, [beta] [greater

than or equal to] 0 and [lambda] [greater than or equal to] 0, [mu] [greater than or equal to] 0 such that [lambda] + [mu] [greater than or equal to] 1; let [{[a.sub.k]}.sup.[infinity].sub.k=1] be any sequence of positive real numbers such that [a.sub.1] = 1, if{[a.sub.k]} satisfy the following conditions:

[mathematical expression not reproducible] (15)

Then [f.sub.n](z) = z + [[summation].sup.n.sub.k=2] [a.sub.k][z.sup.k] and f(z) = z + [[summation].sup.[infinity].sub.k=2] [a.sub.k][z.sup.k] are close-to-convex with respect to starlike function z/(1 - [z.sup.2]).

Note that Theorem 8 provides close-to-convexity of f with respect to the function z/(1 - [z.sup.2]). Results for the close-to-convexity of f with respect to other four starlike functions given in Remark 3 are of considerable interest, and the authors have considered some of these cases separately elsewhere. The next result provides the coefficient conditions for f to be in the class of prestarlike functions of order y, 0 [less than or equal to] [gamma] < 1.

Theorem 9. Let [alpha] [greater than or equal to] 0, [beta] [greater than or equal to] 0, [lambda] [greater than or equal to] 0, [mu] [greater than or equal to] 0 such that [lambda] + [mu] [greater than or equal to] 1; let [{[a.sub.k]}.sup.[infinity].sub.k=1] be any sequence of positive real numbers such that [a.sub.1] = 1. Let {[a.sub.k]} satisfy the following conditions:

(1) [(2 + a).sup.[lambda] [(2 + [beta]).sup.[mu]] (3 - [gamma])(3 - 2[gamma])[a.sub.3] [less than or equal to] 2(2 - [gamma])[a.sub.2] [less than or equal to] [a.sub.1].

(2) [(k + 1 + [alpha]).sup.[lambda]] [(k + 1 + [beta]).sup.[mu]] (k + 2 - [gamma])(k + 2 - 2[gamma])[a.sub.k+2] [less than or equal to] [(k + [alpha]).sup.[lambda]] [(k + [beta]).sup.[mu]] (k + 1 - [gamma])(k + 1)[a.sub.k+1], [for all]k [greater than or equal to] 2.

Then for 0 [less than or equal to] [gamma] < 1, [f.sub.n](z) = z + [[summation].sup.n.sub.k=2] [a.sub.k][z.sup.k] is prestarlike of order [gamma]. Moreover, f(z) = z + [[summation].sup.n.sub.k=2] [a.sub.k][z.sup.k] is prestarlike of order [gamma].

Proof. Let [g.sub.n](z) := [f.sub.n](z) * z/[(1 - z).sup.2-2[gamma]], z [member of] D, 0 [less than or equal to] [gamma] < 1. To prove required theorem, it is sufficient to prove that [g.sub.n](z) [member of] [S.sup.*]([gamma]):

[g.sub.n](z) = [f.sub.n](z) * [z/[(1 - z).sup.2-2[gamma]]]

= z + [[summation].sup.n.sub.k=2] [[(2 - 2[gamma]).sub.k-1]/(k - 1)!] [a.sub.k][z.sup.k], z [member of] D. (16)

We prove that [g.sub.n](z) satisfy the conditions of Lemma 4. For this, define

[h.sub.n](z) := [g'.sub.n](z) - [gamma][[g.sub.n](z)/z] = [[b.sub.0]/2] + [n-1.summation over (k=1)], z [member of] D, (17)

where [b.sub.0] = 2(1 - [gamma]) and [b.sub.k] = (k + 1 - [gamma])[((2 - 2[gamma]).sub.k]/k!)[a.sub.k+1] for k [greater than or equal to] 1. Using simple calculations, along with the hypothesis, {[b.sub.k]} satisfy the conditions of Corollary 2. Continuing the same argument as earlier, we get the desired result.

Remark 10. Note that R(1/2) = [S.sup.*](1/2). It can be easily verified that all the conditions of Theorem 9 for R(1/2) coincide with the conditions of Theorem 5 for [S.sup.*](1/2).

For [gamma] = 0, [R.sup.*](0) = C and the following result is immediate.

Corollary 11. For [alpha] [greater than or equal to] 0, [beta] [greater than or equal to] 0, [lambda] [greater than or equal to] 0, [mu] [greater than or equal to] 0 such that [lambda] + [mu] [greater than or equal to] 1, let [{[a.sub.k]}.sup.[infinity].sub.k=1] be any sequence of positive real numbers such that [a.sub.1] = 1. Let {[a.sub.k]} satisfy the following condition:

[mathematical expression not reproducible] (18)

Then [f.sub.n](z) = z + [[summation].sup.n.sub.k=2][a.sub.x][z.sup.k] is convex function. In particular f(z) = z + [[summation].sup.[infinity].sub.k=2] is convex univalent.

Example 12. Let f(z) = z + [z.sup.2]/4 + [[summation].sup.[infinity].sub.k=3] ([z.sup.k]/[(k - 1 + [alpha]).sup.[lambda]] [(k - 1 + [beta]).sup.[mu]][k.sup.2]) is convex univalent.

In particular if [alpha] = [beta] = 1 and [lambda] = [mu] = 1/2, we get that z + [z.sup.2]/4 + [[summation].sup.[infinity].sub.k=3]([z.sup.k]/[k.sup.3]) is convex.

3. Application to Cesaro Mean of Type (b - 1, c)

The nth Cesaro mean of type (b - 1, c) of f(z) [member of] [A.sub.0] is given by

[s.sup.(b-1,c).sub.n](z, f) := z [n.summation over (k=2)][[B.sub.n-k]/[B.sub.n-1]][a.sub.k][z.sup.k] = [s.sup.(b-1,c).sub.n](z) * f(z), n [member of] N, (19)

where b and c are real numbers such that b + 1 > c > 0 and [B.sub.0] = 1 and [B.sub.k] = ((1 + b - c)/b)([(b).sub.k]/[(c).sub.k]) for k [greater than or equal to] 1. Here by [([alpha]).sub.k], k [greater than or equal to] N, which is the well-known Pochhammer symbol, we mean the following:

[([alpha]).sub.k] = [alpha][([alpha] + 1).sub.k-1] with [([alpha]).sub.0] = 1. (20)

For b = 1 + [delta] and c = 1, it follows that

[s.sup.(1).sub.n](f, z) = [s.sup.[delta].sub.n](f,z) = z + [n.summation over (k=2)][[(1 + [delta]).sub.n-k]/(n - k)!] [(n - 1)/[(1 + [delta]).sub.n-1]][a.sub.k][z.sup.k], (21)

which is the Cesaro mean of order [delta] for [delta] > -1. Since (19) is one type of generalization of the well-known Cesaro mean , we call these Cesaro means of type (b - 1; c) as generalized Cesaro operators. The coefficients given in (19) were considered in  while finding positivity of trigonometric polynomials. Using (19), generalized Cesaro averaging operators were studied in  which are generalization of the Cesaro operator given by Stempak . The geometric properties of [s.sup.[delta].sub.n](z) are well-known. For details, see [23, 26, 27]. Lewis  proved that [s.sup.[delta].sub.n](z) is close-to-convexand hence univalent for [delta] [greater than or equal to] 1. Ruscheweyh  proved that it is prestarlike of order (3 - [delta])/2. Hence it would be interesting to see if the geometric properties of [s.sup.[delta].sub.n](z) can be extended to [s.sup.b-1,c.sub.n](f, z). Such investigations are possible by various known methods in geometric function theory. In particular, the positivity techniques used in Koumandos  or Mondal and Swaminathan  can be applied to [s.sup.b-1,c.sub.n](z) as well. However, in view of Example 7, we are interested in using the results available in Section 2 to obtain the geometric properties of [s.sup.b-1,c.sub.n](z).

Theorem 13. Let {[a.sub.k]} be any sequence of positive real numbers such that [a.sub.1] = 1 and (b + n - 2)[a.sub.1] [greater than or equal to] 2(c + n -2)[a.sub.2]. Let b [greater than or equal to] c > 0, 0 [less than or equal to] [alpha] [less than or equal to] 6/([lambda] + 4), 0 [less than or equal to] [beta] [less than or equal to] 6/([mu] + 4), and [lambda], [mu] [greater than or equal to] 0 such that [lambda] + [mu] [greater than or equal to] 1 and 1 [less than or equal to] [lambda] + [mu] < 2 and satisfy the following conditions:

(i) (2 - [alpha][lambda])(2 - [beta][mu])(b + n - 3)[a.sub.2] [greater than or equal to] [2.sup.[lambda]+[mu]+1](c + n - 3)[3.sub.a3].

(ii) (k - 1 + [alpha] - [lambda])(k - 1 + [beta] - [mu])(b + n - k - 1)k[a.sub.k] [greater than or equal to] (k - 1 + [alpha])(k - 1 + [beta])(c + n - k - 1)(k + 1)[a.sub.k+1] for 3 [less than or equal to] k [less than or equal to] n - 3.

(iii) (n - 2 + [alpha] - [lambda])(n - 2 + [beta] - [mu])(1 + b - c)(n - 1)[a.sub.n-1] [greater than or equal to] (n - 2 + [alpha])(n - 2 + [beta])cn[a.sub.n].

Then [s.sup.b-1,c.sub.n](f, z) is close-to-convex with respect to z and z/(1 - z) where f(z) = z + [[summation].sup.[infinity].sub.k=2] [a.sub.k][z.sup.k]. Further for the same condition [s.sup.b-1,c.sub.n](f, z) is starlike univalent.

Proof. Let [s.sup.b-1,c.sub.n](f, z) = z + [[summation].sup.n.sub.k=2] ([B.sub.n-k]/[B.sub.n-1])[a.sub.k][z.sup.k]. Then,

[s.sup.b-1,c.sub.n](f, z)' = 1 + [n-1.summation over (k=1)][[B.sub.n-k-1]/[B.sub.n-1]](k + 1)[a.sub.k+1][z.sup.k]. (22)

For 0 [less than or equal to] r < 1 and 0 [less than or equal to] [theta] [less than or equal to] 2[pi],

[mathematical expression not reproducible] (23)

where [b.sub.0] = 2 and = [b.sub.k]([B.sub.n-k-1]/[B.sub.n-1])(k + 1)[a.sub.k+1] for k [greater than or equal to] 1. Hence [b.sub.k] and [b.sub.k+1] can be related as follows:

[mathematical expression not reproducible] (24)

For the sequence {[b.sub.k]}, our aim is to prove that [b.sub.0]/2 + [[summation].sup.n-1.sub.k=1][b.sub.k][r.sup.k] cos k[theta] > 0 and [[summation].sup.n-1.sub.k=1] [b.sub.k][r.sup.k] sin k[theta] > 0. Note that

[[b.sub.0]/2] - [b.sub.1]

= [1/(b + n - 2)][(b + n - 2)[a.sub.1] - 2(c + n - 2)[a.sub.2]]

> 0. (25)

For a given [alpha] and [beta], we can easily get

[mathematical expression not reproducible] (26)

Hence we see that

[mathematical expression not reproducible] (27)

For the other condition [(k + [alpha]).sup.[lambda]][(k + [beta]).sup.[mu]] [b.sub.k] [greater than or equal to] [(k + 1 + [alpha]).sup.[lambda]] [(k + 1 + [beta]).sup.[mu]][b.sub.k+1] to be satisfied, first we find

[mathematical expression not reproducible] (28)

Clearly,

[mathematical expression not reproducible] (29)

For k = n - 2, consider

[mathematical expression not reproducible] (30)

We proved that [b.sub.0]/2 + [[summation].sup.n-1.sub.k=1][b.sub.k] cos k[theta] > 0 and [[summation].sup.n-1.sub.k=1] [b.sub.k] sin k[theta] > 0 for 0 < [theta] < [pi]. By the minimum principle for harmonic functions, [b.sub.0]/2 + [[summation].sup.n-1.sub.k=1][b.sub.k] cos k[theta] > 0, 0 [less than or equal to] r < 1 and 0 < [theta] < [pi] and [[summation].sup.n-1.sub.k=1] [b.sub.k] sin k[theta] > 0 for 0 < [theta] < [pi] and 0 [less than or equal to] r < 1. Using reflection principle, [[summation].sup.n-1.sub.k=1][b.sub.k][r.sup.k] sin k[theta] < 0 for [pi] < [theta] < 2[pi] and 0 [less than or equal to] r < 1. Note that [s.sup.(b-1,c).sub.n] f, z) is close-to-convex with respect to z if Re [s.sup.(b-1,c).sub.n](f, z) > 0 and [s.sup.(b-1,c).sub.n)](f,z) is close-to-convex with respect to z/(1 - z) if Re[(1 - z) [s.sup.(b-1,c).sub.n)](f, z)'] > 0. Now

[mathematical expression not reproducible] (31)

For b = 1 + [delta], c = 1, Theorem 13 leads to the following example.

Example 14. Let [lambda] [greater than or equal to] 0, [mu] [greater than or equal to] 0 such that 1 [less than or equal to] [lambda] + [mu] < 2, 0 [less than or equal to] [alpha] [less than or equal to] 6/([lambda] + 4), and 0 [less than or equal to] [beta] [less than or equal to] 6/([beta] + 4); then

[mathematical expression not reproducible] (32)

Then [s.sup.[delta].sub.n](-log(1 - z), z) is close-to-convex with respect to z and z/(1-z). Further for the same condition it is also starlike univalent.

Remark 15. If we take [alpha] = [beta] = 1 and [lambda] = [mu] = 1/2 then, for 1 [less than or equal to] n [less than or equal to] 3, [s.sup.[delta].sub.n](- log(1 - z), z) is close-to-convex with respect to z and z/(1 - z) for [delta] [greater than or equal to] [delta]' where 0 < [delta]' < 3. This conclusion cannot be obtained from [20, Corollary 4.2].

Theorem 16. Let {[a.sub.k]} be a sequence of positive real numbers with [a.sub.1] = 1 and satisfy the hypothesis of Theorem 13. Then [s.sup.(b-1,c).sub.n)](f,z) [member of] R([gamma]), [gamma] [greater than or equal to] 0 where

[gamma] [less than or equal to] 1 - [(c + n - 2)/(b + n - 2)] 2[a.sub.2], (33)

R([gamma]) = {f [member of] A : Re f'(z) > [gamma]] and f(z) = z + [[summation].sup.[infinity].sub.k=2] [a.sub.k][z.sup.k], z [member of] D.

Proof. Let [s.sup.(b-1,c).sub.n)](f,z) = z + [[summation].sup.n.sub.k=2]([B.sub.n-k]/[B.sub.n-1])[a.sub.k][z.sup.k] where [B.sub.0] = 1 and [B.sub.k] = ([(b).sub.k]/[(c).sub.k])((1 + b - c)/b), k [greater than or equal to] 1:

[s.sup.(b-1,c).sub.n)](f, z)' = 1 + [n-1.summation over (k=1)][[B.sub.n-k-1]/[B.sub.n-1]](k + 1) [a.sub.k+1][z.sup.k]. (34)

We consider

[[s.sup.(b-1,c).sub.n)](f,z)' - [gamma]/1 - [gamma]] = [[b.sub.0]/2] + [n-1.summation over (k=1)][b.sub.k][z.sup.k], (35)

where [b.sub.0] = 2 and [b.sub.k] = [B.sub.n-k-1]/[B.sub.n-1] x (k + 1)[a.sub.k+1]/(1 - [gamma]) for k [greater than or equal to] 1. Then [b.sub.k] and [b.sub.k+1] are related by

[b.sub.k+1] = (c + n - k - 2)(k + 2) [a.sub.k+2]/(b + n - k - 2) (k + 1) [a.sub.k+1] (36)

for 1 [less than or equal to] k [less than or equal to] n - 3,

and, for k = n - 2,

[b.sub.n-1] = [(1 + b - c)/c][(n - 1)[a.sub.n-1]/n[a.sub.n]][b.sub.n-2]. (37)

Using hypothesis, we can easily get

[[b.sub.0]/2] - [b.sub.1] = 1 - (c + n - 2/b + n - 2) [2[a.sub.2]/1 - [gamma]] [greater than or equal to] 0. (38)

The relation between the coefficients [b.sub.k] and [b.sub.k+1] is the same as in the Theorem 13. So such [b.sub.k] also satisfy the conditions of Theorem 13 and from Corollary 2 we have the required result that

[[b.sub.0]/2] + [n-1.summation over (k=1)][b.sub.k] cos k[theta] > 0 for 0 < [theta] < [pi]. (39)

From the minimum principle for harmonic functions for 0 [less than or equal to] r < 1 and 0 < [theta] < 2[pi] we have

Re([s.sup.(b-1,c).sub.n)](f,z) - [gamma]/1 - [gamma]) = [[b.sub.0]/2] + [n-1.summation over (k=1)][b.sub.k][r.sup.k] cos k[theta] > 0. (40)

So, [s.sup.(b-1,c).sub.n)](f,z) [member of] R([gamma]).

It can be clearly seen that, for [gamma] = 0, Theorem 16 coincides with Theorem 13 for the case g(z) = z.

Theorem 17. Let [{[a.sub.k]}.sup.[infinity].sub.k=1] be a sequence of positive real numbers such that [a.sub.1] = 1. If for [lambda] [greater than or equal to] 0, [mu] [greater than or equal to] 0 such that 1 [less than or equal to] [lambda] + [mu] < 2 and 0 [less than or equal to] [alpha] [less than or equal to] 6/([lambda] + 4), 0 [less than or equal to] [beta] [less than or equal to] 6/([mu] + 4), [a.sub.k] satisfy the following conditions:

(1) (3 - 2[lambda] - 2[mu])(b + n - 2)[a.sub.1] [greater than or equal to] (5 - 2[lambda] - 2[mu])(c + n - 2)[a.sub.2].

(2) (2 - [alpha][lambda])(2 - [beta][mu])(5 - 2[lambda] - 2[mu])(b + n - 3)[a.sub.2] [greater than or equal to] [2.sup.[lambda]+[mu]+2](7 - 2[lambda] - 2[mu])(c + n - 3)[a.sub.3].

(3) (2k + 1 - 2[lambda] - 2[mu])(k -1 + [alpha] - [lambda])(k - 1 + [beta] - [mu])(b + n - k - 1)[a.sub.k] [greater than or equal to] (2k + 3 - 2[lambda] - 2[mu])(k - 1 + [alpha])(k - 1 + [beta])(c + n - k - 1)[a.sub.k+1] for 3 [less than or equal to] k [less than or equal to] n - 2.

(4) (n - 2 + [alpha] - [lambda])(n - 2 + [beta] - [mu])(2n + 1 - 2[lambda] - 2[mu])(1 + b - c)[a.sub.n-1] [greater than or equal to] (n - 2 + [alpha])(n - 2 + [beta])(2n + 3 - 2[lambda] - 2[mu])c[a.sub.n].

Then, [s.sup.(b-1,c).sub.n](f,z) [member of] [S.sup.*]([lambda] + [mu] - 1/2), where f(z) = z + [[summation].sup.[infinity].sub.k=2][a.sub.k][z.sup.k], z [member of] D.

Proof. [s.sup.(b-1,c).sub.n](f,z) = z + [[summation].sup.n.sub.k=2]([B.sub.n-k]/[B.sub.n-1])[a.sub.k][z.sup.k] = [b.sub.1]z + [[summation].sup.n.sub.k=2][b.sub.k][z.sup.k], where [b.sub.1] = 1 and [b.sub.k] = ([B.sub.n-k]/[B.sub.n-1])[a.sub.k] for k [greater than or equal to] 2. Then,

[b.sub.k+1] = (c + n - k - 1/b + n - k - 1) [[a.sub.k+1]/[a.sub.k]][b.sub.k], for 2 [less than or equal to] k [less than or equal to] n - 2, (41)

and for k = n - 1, [b.sub.n] = (c/(1 + b - c))([a.sub.n]/[a.sub.n-1])[b.sub.n-1]. It is enough to prove that {[b.sub.k]} satisfy the conditions of Theorem 5. For the sake of convenience we substitute [gamma] = [lambda] + [mu] - 1/2. By a simple calculation we can get that (1 - [gamma])[b.sub.1] - (2 - [gamma])[b.sub.2] [greater than or equal to] 0. Now

[mathematical expression not reproducible] (42)

Now, for 2 [less than or equal to] k [less than or equal to] n - 3,

[mathematical expression not reproducible] (43)

And, for k = n - 2, using the hypothesis, we obtain,

[mathematical expression not reproducible] (44)

From Theorem 5 the desired result follows.

Theorem 18. Let b [greater than or equal to] c > 0, 0 [less than or equal to] [alpha] [less than or equal to] 6/([lambda] + 4), 0 [less than or equal to] [beta] [less than or equal to] 6/([mu] + 4) and [lambda], [mu] [greater than or equal to] 0 such that 1 [less than or equal to] [lambda] + [mu] < 2 and satisfies the following conditions:

(1) 2(2 - [gamma])(c + n - 2) [less than or equal to] (b + n - 2),

(2) [(2 + [alpha]).sup.[lambda]][(2 + [beta]).sup.[mu]](3 - [gamma])(3 - 2[gamma])(c + n - 3) [less than or equal to] 2(2 - [gamma])(b + n - 3),

(3) [(k + 1 + [alpha]).sup.[lambda]][(k + 1 + [beta]).sup.[mu]](k + 2 - [gamma])(k + 2 - 2[gamma])(c + n - k - 2) [less than or equal to] [(k + [alpha]).sup.[lambda]][(k + [beta]).sup.[mu]] (k + 1 - [gamma])(k + 1)(b + n - k - 2) for 2 [less than or equal to] k [less than or equal to] n - 3,

(4) [(n - 1 + [alpha]).sup.[lambda]] [(n - 1 + [beta]).sup.[mu]](n - [gamma])(n - 2[gamma])c [less than or equal to] [(n - 2 + [alpha]).sup.[lambda]][(n - 2 + [beta]).sup.[mu]](n - 1 - [gamma])(n - 1)(1 + b - c).

Then [s.sup.(b-1,c).sub.n](z) is prestarlike of order [gamma], where 0 [less than or equal to] [gamma] < 1.

Proof. It is given that [s.sup.(b-1,c).sub.n](z) = z + [[summation].sup.n.sub.k=2]([B.sub.n-k]/[B.sub.n-1])[z.sup.k] = z + [[summation].sup.n.sub.k=2][a.sub.k][z.sup.k], z [member of] D.

Then using [a.sub.k] = [B.sub.n-k]/[B.sub.n-1] for k [greater than or equal to] 1 in Theorem 5 and following the same procedure the result can be proved.

If [gamma] = 0 then [s.sup.(b-1,c).sub.n](z) [member of] [R.sup.*](0) = C. Further if we substitute b = 1 + [delta] and c = 1 in Theorem 18, we have the following example.

Example 19. If [alpha], [beta], [lambda], and [mu] satisfy the conditions of Theorem 18 and if

[mathematical expression not reproducible] (45)

Then [s.sup.[delta].sub.n](z) is prestarlike of order [gamma], where [gamma] [member of] [0,1).

It can be noted that if we take [alpha] = [beta] = 0 and [lambda] + [mu] = 1 in Example 19, then, for [delta] [greater than or equal to] (n - 1)(3 - 2[gamma]), [s.sup.[delta].sub.n](z) [member of] [R.sup.*]([gamma]). Similar type of result had been found in [23, Theorem 1]. From [26, Theorem 2.1], we deduce the following corollary.

Corollary 20. If [alpha], [beta], [lambda], [mu], and [gamma] satisfy the hypothesis of Theorem 18 then for b [greater than or equal to] c, [s.sup.(b-1,c).sub.n] [R.sup.*]([gamma]). Then for any zg [member of] K([gamma]) [??] g * ([s.sup.(b-1,c).sub.n](z))' is zero free in D.

The following Lemma, which is the extension of the well-known Polya-Schoenberg Theorem, is ingredient to our next result.

Lemma 21 (see [17, p. 499]). If f [member of] K([gamma]), g [member of] [R.sup.*]([gamma]), 0 [less than or equal to] [gamma] < 1 then f * g [member of] K([gamma]).

Clearly, [s.sup.(b-1,c).sub.n](g, z) [member of] K([gamma]) if g [member of] K([gamma]).

Theorem 22. Let {[a.sub.k]} be a sequence of positive real numbers such that [a.sub.1] = 1. Then, for 0 [less than or equal to] [alpha] [less than or equal to] 6/([lambda] + 4), 0 [less than or equal to] [beta] [less than or equal to] 6/([mu] + 4) and [lambda] [greater than or equal to] 0, [mu] [greater than or equal to] 0 such that 1 [less than or equal to] [lambda] + [mu] < 2, if {[a.sub.k]} satisfy the following conditions:

(1) (2 - [alpha][lambda])(2 - [beta][mu])(b + n - 2)[a.sub.1] [greater than or equal to] (c + n - 2)[2.sup.[lambda]+[mu]+3][a.sub.2].

(2) k(k + [alpha] - [lambda])(k + [beta] - [mu])(b + n - k - 1)[a.sub.k] [greater than or equal to] (k + [alpha])(k + [beta])(c + n - k - 1)(k + 1)[a.sub.k+1], for all 2 [less than or equal to] k [less than or equal to] n - 2.

(3) (n - 1 + [alpha] - [lambda])(n - 1 + [beta] - [mu])(1 + b - c)(n - 1)[a.sub.n-1] [greater than or equal to] c(n - 1 + [alpha])(n - 1 + [beta])n[a.sub.n].

Then [s.sup.(b-1,c).sub.n](f, z) is close-to-convex with respect to starlike function z/(1 - [z.sup.2]) where f(z) = z + [[summation].sup.[infinity].sub.k=0][a.sub.k][z.sup.k], z [member of] D.

Proof. [s.sup.(b-1,c).sub.n](f,z) = z + [[summation].sup.n.sub.k=2]([B.sub.n-k]/[B.sub.n-1])[a.sub.k][z.sup.k] is close-to-convex with respect to z/(1 - [z.sup.2]) if z[s.sup.(b-1,c).sub.n](f,z)' is typically real function. Consider

z[s.sup.(b-1,c).sub.n](f, z)' = z + [[summation].sup.n.sub.k=2] [[B.sub.n-k]/[B.sub.n-1]]k[a.sub.k][z.sup.k] = [b.sub.1]z + [[summation].sup.n.sub.k=2][b.sub.k][z.sup.k], (46)

where [b.sub.1] = 1 and [b.sub.k] = ([B.sub.n-k]/[B.sub.n-1])k[a.sub.k] for k [greater than or equal to] 2. Clearly

[mathematical expression not reproducible] (47)

Now,

[mathematical expression not reproducible] (48)

Further, for 2 [less than or equal to] k [less than or equal to] n - 2,

[mathematical expression not reproducible] (49)

For k = n - 1,

[mathematical expression not reproducible] (50)

which is nonnegative. Following the same argument as in Theorem 5, z[s.sup.(b-1,c).sub.n](f, z) is typically real which completes the proof.

Remark 23. Note that we have no result for the close-to-convexity of [s.sup.(b-1,c).sub.n](f, z) with respect to the starlike functions z/[(1 - z).sup.2] and z/(1 - z + [z.sup.2]). Although there are not many results in the literature for close-to-convexity with respect to z/(1 - z + [z.sup.2]), it will be interesting if one can find the results in this direction.

https://doi.org/10.1155/2017/6584584

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The first author acknowledges the Council of Scientific and Industrial Research, India (Grant code 09/143(0827)/2013-EMR-1), for financial support to carry out the above research work.

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Priyanka Sangal and A. Swaminathan

Department of Mathematics, Indian Institute of Technology (IIT) Roorkee, Uttarakhand 247667, India

Correspondence should be addressed to Priyanka Sangal; sangal.priyanka@gmail.com

Received 10 August 2017; Accepted 9 November 2017; Published 28 November 2017

Academic Editor: Pranay Goswami
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