Geometric properties of Cesaro averaging operators

In this paper, using positivity of trigonometric cosine and sine sums whose coefficients are generalization of Vietoris numbers, we find the conditions on the coefficient $\{a_k\}$ to characterize the geometric properties of the corresponding analytic function $f(z)=z+\displaystyle\sum_{k=2}^{\infty} a_kz^k$ in the unit disc $\mathbb{D}$. As an application we also find geometric properties of a generalized Ces\`aro type polynomials.


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 [1] and Ruscheweyh and Salinas [2]. For a detailed application in signal processing, we refer to the monograph of Dumitrescu [3]. For other applications in this direction, we refer to Dimitrov and Merlo [4], Fernández-Durán [5], and Gasper [6]. The positive trigonometric polynomials played an important role in the proof of Bieberbach conjecture; see [7]. For the applications of positive trigonometric polynomials in Fourier series, approximation theory, function theory, and number theory, we refer to the work of Dimitrov [8] and references therein. For the study of extremal problems, we refer to the dissertation of Révész [9] 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 [10] followed by Koumandos [11] are of interest to the present investigation. Precisely, Vietoris [10] gave sufficient conditions on the coefficient of a general class of sine and cosine sums that ensure their positivity in (0, ). For further details in this direction one can refer to [11][12][13] and the references therein. An account of recent results available in this direction is given in [13] and one of the main results in [13] is as follows.
Using summation by parts, the following corollary of Theorem 1 can be obtained. then, for ∈ N, the following inequalities hold: 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 0 be the subclass of the class of analytic functions ∈ A with normalized conditions (0) = 0, (0) = 1 in the unit disc D = { ∈ D, | | < 1}. The subclasses of A 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 ≤ < 1, let S * ( ) be the family of functions starlike of order ; that is, if ∈ A 0 satisfies the analytic characterization, For 0 ≤ < 1, let ( ) be the family of functions convex of order ; that is, if ∈ A 0 satisfies the analytic characterization, These two classes are related by the Alexander transform, ∈ C( ) ⇔ ∈ S * ( ). The usual classes of starlike functions (with respect to origin) and convex functions are denoted, respectively, by S * (0) ≡ S * and C(0) ≡ C. An analytic function is said to be close-to-convex of order , (0 ≤ < 1) with respect to a fixed starlike function if and only if it satisfies the analytic characterization: The family of all close-to-convex function of order with respect to ∈ S * is denoted by K ( ). Further, for 0 ≤ < 1, each ∈ K ( ) is also univalent in D. The proper inclusion between these classes is given by Another important subclass is the class of typically real functions. A function ∈ A 0 is typically real if Im( )Im( ( )) ≥ 0 where ∈ 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
are the only nine starlike univalent functions having integer coefficients in D. It will be interesting to find to be close-toconvex when the corresponding starlike function takes one of the above forms.
If we take = 0 and ( ) = /(1 − ) 2 then Re((1 − ) 2 ( )) > 0 which implies ( ) is typically real function. A function ∈ A 0 is said to be typically real if Im ( )Im( ) > 0 whenever Im( ) ̸ = 0, ∈ D. The function ( ) fl /(1 − ) 2−2 is the extremal function for the class of starlike function of order . Note that 0 ( ) is the wellknown Koebe function and the function 1/2 ( ) = /(1 − ) is the extremal function for the class C. A function ( ) is said to be prestarlike of order , 0 ≤ < 1, if ( ) * ( ) = /(1 − ) 2 * ( ) ∈ S * ( ) where " * " is the convolution operator or Hadamard product. This class was introduced by Ruscheweyh [17]. For more details of this class see [18]. Here the Hadamard product or convolution is defined as follows: 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 [19] obtained some coefficient conditions for the class of starlike functions using the classical result of Vietoris [10]. So it would be interesting to find the geometric properties of function ( ) in which Corollary 2 plays a vital role.

Geometric Properties of an Analytic Function
In this section, we provide conditions on the Taylor coefficients of an analytic function to guarantee the admissibility of 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.
By proving that ( ) is typically real function in the similar fashion, we obtain the next result.
be any sequence of positive real numbers such that 1 = 1, if { } satisfy the following conditions: Note that Theorem 8 provides close-to-convexity of with respect to the function /(1 − 2 ). Results for the close-to-convexity of 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 to be in the class of prestarlike functions of order , 0 ≤ < 1.
For = 0, R * (0) ≡ C and the following result is immediate.

Application to Cesàro Mean of Type ( − 1, )
The th Cesàro mean of type ( − 1, ) of ( ) ∈ A 0 is given by where and are real numbers such that + 1 > > 0 and 0 = 1 and = ((1 + − )/ )(( ) /( ) ) for ≥ 1. Here by ( ) , ≥ N, which is the well-known Pochhammer symbol, we mean the following: For = 1 + and = 1, it follows that ( ,1) ( , ) = ( , ) which is the Cesàro mean of order for > −1. Since (19) is one type of generalization of the well-known Cesàro mean [23], we call these Cesàro means of type ( − 1; ) as generalized Cesàro operators. The coefficients given in (19) were considered in [13] while finding positivity of trigonometric polynomials. Using (19), generalized Cesàro averaging operators were studied in [24] which are generalization of the Cesàro operator given by Stempak [25]. The geometric properties of ( ) are well-known. For details, see [23,26,27]. Lewis [28] proved that ( ) is close-to-convex and hence univalent for ≥ 1. Ruscheweyh [23] proved that it is prestarlike of order (3 − )/2. Hence it would be interesting to see if the geometric properties of ( ) can be extended to ( −1, ) ( , ). Such investigations are possible by various known methods in geometric function theory. In particular, the positivity techniques used in Koumandos [11] or Mondal and Swaminathan [20] can be applied to ( −1, ) ( ) 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 ( −1, ) ( ).
It can be clearly seen that, for = 0, Theorem 16 coincides with Theorem 13 for the case ( ) = .
which is nonnegative. Following the same argument as in Theorem 5, ( −1, ) ( , ) is typically real which completes the proof.
Remark 23. Note that we have no result for the close-toconvexity of ( −1, ) ( , ) with respect to the starlike functions /(1 − ) 2 and /(1 − + 2 ). Although there are not many results in the literature for close-to-convexity with respect to /(1 − + 2 ), it will be interesting if one can find the results in this direction.

Conflicts of Interest
The authors declare that they have no conflicts of interest.