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Geometric Properties of Cesàro Averaging Operators
Using positivity of trigonometric cosine and sine sums whose coefficients are generalization of Vietoris numbers, we find the conditions on coefficient to characterize the geometric properties of the corresponding analytic function in the unit disc . As an application, we also find geometric properties of generalized Cesàro-type polynomials.
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 , Fernández-Durán , 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 Révész  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 . 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 , , and such that then for , , and , , we have for and .
Using summation by parts, the following corollary of Theorem 1 can be obtained.
Corollary 2. For , and , such that . If there exists a sequence of positive real numbers, such that then, for , 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 be the subclass of the class of analytic functions with normalized conditions , in the unit disc . The subclasses of consisting of univalent function are denoted by . Several subclasses of univalent functions play a prominent role in the theory of univalent functions. For , let be the family of functions starlike of order ; that is, if satisfies the analytic characterization, For , let be the family of functions convex of order ; that is, if satisfies the analytic characterization, These two classes are related by the Alexander transform, . The usual classes of starlike functions (with respect to origin) and convex functions are denoted, respectively, by and . An analytic function is said to be close-to-convex of order , 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 is denoted by . Further, for , each is also univalent in . The proper inclusion between these classes is given by Another important subclass is the class of typically real functions. A function is typically real if where . Its class is denoted by . 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 . It will be interesting to find to be close-to-convex when the corresponding starlike function takes one of the above forms.
If we take and then which implies is typically real function. A function is said to be typically real if whenever , . The function is the extremal function for the class of starlike function of order . Note that is the well-known Koebe function and the function is the extremal function for the class . A function is said to be prestarlike of order , , if 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 and , . Then,
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 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 to guarantee the admissibility of in subclasses of , using Corollary 2. The next lemma which is the generalization of [19, Lemma ] is the crucial ingredient in the proof of the following theorem.
Lemma 4 (see [20, Theorem ]). Let and be such that and are typically real in . Further if and , then .
Theorem 5. Let , , , such that ; let be any sequence of positive real numbers such that . Let satisfy the following conditions: (1).(2).(3), . Then, for , and are starlike of order .
Proof. Let , be the partial sum of . Then .
Define where and , . Consider, Now, for , So by the given hypothesis, satisfy the conditions of Corollary 2 which implies and if . By reflection principle if . So is typically real function. In order to prove the theorem it is remaining to show that and is typically real. In this case and . So such also satisfy given hypothesis because , for all . So and again using reflection principle we get that is typically real in .
Applying Lemma 4, we get that . Since and the family of starlike functions is normal [21, p. ], we get is also starlike of order .
Remark 6. If in Theorem 5, then we get which implies is close-to-convex with respect to and is typically real also and with this yields that is close-to-convex with respect to starlike function .
Example 7. Consider the sequence as , , and for ; then by Theorem 5, the function is starlike univalent. But [22, Theorem ] fails to include this function. Hence Theorem 5 is better than [22, Theorem ] in the sense that it is likely to include more cases.
By proving that is typically real function in the similar fashion, we obtain the next result.
Theorem 8. Let , and , such that ; let be any sequence of positive real numbers such that , if satisfy the following conditions: Then and are close-to-convex with respect to starlike function .
Note that Theorem 8 provides close-to-convexity of with respect to the function . 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 , .
Theorem 9. Let , , , such that ; let be any sequence of positive real numbers such that . Let satisfy the following conditions: (1).(2), . Then for , is prestarlike of order . Moreover, is prestarlike of order .
Proof. Let , , . To prove required theorem, it is sufficient to prove that :We prove that satisfy the conditions of Lemma 4. For this, define where and for . Using simple calculations, along with the hypothesis, satisfy the conditions of Corollary 2. Continuing the same argument as earlier, we get the desired result.
Remark 10. Note that . It can be easily verified that all the conditions of Theorem 9 for coincide with the conditions of Theorem 5 for .
For , and the following result is immediate.
Corollary 11. For , , , such that , let be any sequence of positive real numbers such that . Let satisfy the following condition: Then is convex function. In particular is convex univalent.
Example 12. Let is convex univalent.
In particular if and , we get that is convex.
3. Application to Cesàro Mean of Type
The th Cesàro mean of type of is given by where and are real numbers such that and and for . Here by , , which is the well-known Pochhammer symbol, we mean the following: For and , it follows that which is the Cesàro mean of order for . Since (19) is one type of generalization of the well-known Cesàro mean , we call these Cesàro means of type as generalized Cesàro operators. The coefficients given in (19) were considered in  while finding positivity of trigonometric polynomials. Using (19), generalized Cesàro averaging operators were studied in  which are generalization of the Cesàro operator given by Stempak . The geometric properties of are well-known. For details, see [23, 26, 27]. Lewis  proved that is close-to-convex and hence univalent for . Ruscheweyh  proved that it is prestarlike of order . Hence it would be interesting to see if the geometric properties of can be extended to . 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 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 .
Theorem 13. Let be any sequence of positive real numbers such that and . Let , , , and such that and and satisfy the following conditions: (i).(ii) for .(iii). Then is close-to-convex with respect to and where . Further for the same condition is starlike univalent.
Proof. Let . Then, For and , where and for . Hence and can be related as follows: For the sequence , our aim is to prove that and . Note that For a given and , we can easily get Hence we see that For the other condition to be satisfied, first we find Clearly, For , consider We proved that and for . By the minimum principle for harmonic functions, , and and for and . Using reflection principle, for and . Note that is close-to-convex with respect to if and is close-to-convex with respect to if . Now
For , , Theorem 13 leads to the following example.
Example 14. Let , such that , , and ; then Then is close-to-convex with respect to and . Further for the same condition it is also starlike univalent.
Remark 15. If we take and then, for , is close-to-convex with respect to and for where . This conclusion cannot be obtained from [20, Corollary ].
Theorem 16. Let be a sequence of positive real numbers with and satisfy the hypothesis of Theorem 13. Then , where and , .
Proof. Let where and , : We consider where and for . Then and are related by and, for , Using hypothesis, we can easily get The relation between the coefficients and is the same as in the Theorem 13. So such also satisfy the conditions of Theorem 13 and from Corollary 2 we have the required result that From the minimum principle for harmonic functions for and we have So, .
It can be clearly seen that, for , Theorem 16 coincides with Theorem 13 for the case .
Theorem 17. Let be a sequence of positive real numbers such that . If for , such that and , , satisfy the following conditions: (1).(2).(3) for .(4). Then, , where , .
Proof. , where and for . Then, and for , . It is enough to prove that satisfy the conditions of Theorem 5. For the sake of convenience we substitute . By a simple calculation we can get that . Now Now, for , And, for , using the hypothesis, we obtain, From Theorem 5 the desired result follows.
Theorem 18. Let , , and such that and satisfies the following conditions: (1),(2)