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Second Hankel Determinants for the Class of Typically Real Functions
We discuss the Hankel determinants for typically real functions, that is, analytic functions which satisfy the condition in the unit disk Δ. Main results are concerned with and . The sharp upper and lower bounds are given. In general case, for , the results are not sharp. Moreover, we present some remarks connected with typically real odd functions.
Recently, the Hankel determinant has been studied intensively by many mathematicians. The research was focused on for various classes of univalent functions. The papers by Janteng et al. [3, 4], Lee et al. , Vamshee Krishna and Ramreddy , and Selvaraj and Kumar  are worth mentioning here. Janteng et al. derived the exact bounds of for the classes: of star-like functions (), of convex functions (), and of functions whose derivative has a positive real part (). Lee et al.  investigated the Hankel determinant in the general class of star-like functions with respect to a given function . This class was defined by Ma and Minda in . In particular, Lee et al. obtained the results for the following classes: of star-like functions of order (), of lemniscate star-like functions (; for the definition of , see ), and of strongly star-like functions of order (). Vamshee Krishna and Ramreddy  generalized the result of Janteng et al. They gave the bound of in the class of convex functions of order . Selvaraj and Kumar  proved that the estimate of the second Hankel determinant for the class of close-to-convex functions is the same as that for the class . The question whether this bound is good for the class of all univalent functions has no answer yet. One can find some other results in this direction in [10–14].
Taking different set of parameters and , the research on the Hankel determinant is much more difficult. In  Hayami and Owa discussed for functions satisfying or . On the other hand, Babalola  tried to estimate for , , and . Shanmugam et al.  discussed for the class of -star-like functions defined by Mocanu in .
In particular, if and then is known as a classical functional of Fekete-Szegö. A lot of papers have been devoted to the studies concerning this functional. Because is not related to the subject of this paper, we omit recalling results obtained in this direction.
The majority of results concerning the Hankel determinants were obtained for univalent functions. In this paper we discuss functions which, in general, are not univalent. We focus our investigation on typically real functions.
2. Class and the Hankel Determinants for a Selected Functions in
A function that satisfies the condition for is called a typically real function. Let denote the class of all typically real functions. Robertson  proved that if and only if there exists a probability measure on such that the following formula holds:The coefficients of a function can be written as follows:The functions , , which appear in the above formula, are the well-known Chebyshev polynomials of the second kind.
Since all coefficients of are real we look for the lower and the upper bounds of instead of the bound of . At the beginning, let us look at a few examples.
Example 1. All the functions , , are in . Since , we have for each . Moreover, . This and the Turan identity for Chebyshev polynomials result in for each .
Example 2. For a function having the Taylor series expansion there is for even and for odd . In this case, the function is not univalent; the bound of is much greater than 1, the value of the second Hankel determinant for star-like functions or close-to-convex functions.
Example 3. Every Hankel determinant , , for a function is positive. Namely, .
For a given class , we denote by , , the region of variability of three succeeding coefficients of functions in , that is, the set . As it is seen in (3), the coefficients of typically real functions are the Stieltjes integrals of the Chebyshev polynomials of the second kind with respect to a probability measure. Hence, is the closed convex hull of the curve (see, e.g., ).
Lemma 4. The functional , , attains its extreme values on the boundary of .
Proof. The only critical point of , where , , and , is . But . Since may be positive as well as negative for , (see Examples 1 and 3), it means that the extreme values of are attained on the boundary of .
3. Bounds of in
In  Ma proved so-called generalized Zalcman conjecture for the class :We apply this result to prove the following.
Theorem 5. If then .
Proof. The result of Ma and the triangle inequality result in
This result is sharp; the equality holds for . Furthermore, we can see the following.
Corollary 6. For one has
For our next theorem let us cite two results. First one is the obvious conclusion from the Carathéodory theorem and the Krein-Milman theorem. We assume that is a compact Hausdorff space and
Theorem A (see [22, Thm. 1.40]). If is continuous then the convex hull of is a compact set and it coincides with the set .
In the above, the symbols and stand for the set of probability measures on and the cardinality of the support of , respectively.
It means that is atomic measure having at most steps. More precise information about the relation between the measure and the convex hull is presented in the following theorem. In what follows, means the scalar product of and .
Theorem B (see [22, Thm. 1.49]). Let be continuous. Suppose that there exists a positive integer , such that for each nonzero in the number of solutions of any equation , , is not greater than . Then, for every such that belongs to the boundary of the convex hull of the following statements are true: (1)If then(a), or(b) and .(2)If then(a), or(b) and one of the points and belongs to .
This theorem, in slightly modified version, was published in  as Lemma .
Putting , , and , we can see that any equation of the formis equivalent to , where is a polynomial of degree 3. Hence, (8) has at most 3 solutions. According to Theorem B, the boundary of the convex hull of is determined by atomic measures for which support consists of at most 2 points. Moreover, one of them has to be −1 or 1. We have proved the following.
Lemma 7. The boundary of consists of points that correspond to the following functions:or
Now, we are ready to prove the following.
Theorem 8. For one has
Proof. By Lemma 7, it is enough to take functions given by (9) or (10). Consider the following:
(I) Function (9) has the series expansionHence, , whereFrom it follows that the critical points of are as follows: , , , , , , and . Among these points, only lies inside the set .
If or then functions (9) coincide with from Example 1. If then . In each case . For , function (9) takes the form . Then .
If and we have . It means that the greatest value of for functions given by (9) is equal to 1. The extremal function is(II) For functions (10), is equal to , whereMoreover, . Taking into account the symmetry of the range of variability of , we obtain the same result as above also for functions defined by (10). The extremal function is
4. Bounds of in
The proof of the following theorem is obvious.
Theorem 9. If is odd then
Hence, one has the following.
Corollary 10. For one has
In similar way, as it was done for Lemma 7, one can prove the following.
Lemma 11. The boundary of consists of points that correspond to the following functions:or
Theorem 12. For one has
Proof. By Lemma 11, it suffices to discuss functions given by (20) or (21). Consider the following:
(I) For functions (20), we have and, hence, applying the Turan identity, , whereThe expression in brackets is greater than or equal to −1 for all . Hence,(II) If function is given by (21) then Using the Turan identity, it follows that , whereunder the assumptions , , and .
Let and be fixed. Sincethe critical points of are as follows: , , and . It is easily seen that all these points are in . Therefore,For or , the functions given by (21) have the formOne can show directly from formula (30) thatFor , there ishence,If or then is equal toorrespectively. Without loss of generality, we can assume that . Then, while looking for the minimum value of , we can restrict the research to the first stated above case (since expression (35) is not less than expression (34)).
Transforming (35), we obtainTaking the smallest possible (i.e., ) the second and the forth component of this expression will not increase. The value of the third component does not depend only on ; in fact, it depends on . For this reason, we can take . Combining these facts, it yields thatThe smallest value of the right hand side of this inequality is achieved for . In this case,Combining two parts of the proof we obtain the conclusion of the theorem. Furthermore, the above shows that the extremal functions arewhere .
5. Bounds of , , in
It is easily seen that for any typically real function. By Theorem 9, this estimate is sharp providing that is an odd integer. At the beginning of this section we will prove the following.
Theorem 13. For one has
Proof. The coefficients of the series expansion of function can be written as follows:Hence,Sincewe obtainIn order to prove that the estimate is sharp, let us take the measure for which support satisfies condition . This measure corresponds to the function .
Observe that holds not only for the measure stated above. Namely, the value −1 in (42) is taken also if , where is any positive integer less than or equal to . From this we conclude that the support of the measure has points with weights , , such that .
The weights satisfyIndeed, if the support of consists of points then takes the formUsing trigonometric identities we obtainwhich results in (45).
Connecting (45) and we conclude that is of the formorIt means that for even the support of consists of points, and for even the number of points of the support of is equal to or .
Taking into account and Theorem 13, we obtain the following.
Theorem 14. For one has
Unfortunately, this bound is not sharp. However, the following can be conjectured.
Conjecture 15. For any positive integer , the following estimate holds. Moreover, this bound is sharp for even .
This conjecture is supported by the facts that in the theorems concerning and the extremal functions are of the formfor appropriately taken . The exact bounds of the Hankel determinants for these functions are collected in Table 1. They were obtained numerically.
6. Remarks Concerning in
In class we discuss subclass consisting of the functions which are odd. The definition of this class is For the representation formula, similar to (2), is valid. Namely,Function has the Taylor series expansion
The following inequalities are obvious:equalities hold for .
For a given class , let us denote by , , the set . From (53) it follows that is the closed convex hull of the curve
From Theorem 13 and from the equivalencewe getHence, for odd , we know thatThe equality holds for functions (48) or (49) providing that . Then, connecting the components of these formulae in pairs, we obtainWith help of the argument given in the proof of Theorem 13, we eventually obtain the odd functions for which .
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
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