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The Fekete-Szegö inequality for close-to-convex functions with respect to a certain starlike function dependent on a real parameter
Journal of Inequalities and Applications volume 2014, Article number: 65 (2014)
Given , let , . An analytic standardly normalized function f in is called close-to-convex with respect to if there exists such that
For the class of all close-to-convex functions with respect to , the Fekete-Szegö problem is studied.
The classical problem settled by Fekete and Szegö  is to find for each the maximum value of the coefficient functional
over the class of univalent functions f in the unit disk of the form
By applying the Loewner method they proved that
The problem of calculating for various compact subclasses ℱ of the class of all analytic functions f in of the form (1.1), as well as for λ being an arbitrary real or complex number, was considered by many authors (see, e.g., [2–10]).
Let denote the class of starlike functions, i.e., the class of all functions such that
Given and , let denote the class of functions called close-to-convex with argument δ with respect to g, i.e., the class of all functions such that
denote the classes of functions called close-to-convex with respect to g and close-to-convex with argument δ, respectively, and let
By using a specific starlike function g, inequality (1.3) defines the related class . Given , let
It is easy to check that each satisfies (1.2), i.e., for every . Then (1.3) is of the form
and defines the class , and further the class . Such classes of functions were studied in [14, 15] and , where some generalization of the Robertson condition for convexity in one direction  was discussed.
Note that for we get the Koebe function . Then condition (1.5) defines the class and further the class of functions close-to-convex with respect to the Koebe function. Such functions have a well-known geometrical meaning, namely condition (1.5) geometrically says that the function has the boundary normalization
and is a domain such that for every . Such functions h, clearly univalent as close-to-convex, and domains are called convex in the positive (negative) direction of the real axis and are related to functions convex in the direction of the imaginary axis (see, e.g., [17–20], [, Chapter VI], ).
For we have the identity function , , and then condition (1.5) is of the form
Functions f having such a property are called of bounded turning with argument δ and form the class denoted usually as . Functions in the class are usually called of bounded turning (cf. [, Vol. I, p.101]). On the other hand, condition (1.6) is known as the famous criterium of univalence due to Noshiro  and Warschawski  (cf. [, Vol. I, p.88]). In this way condition (1.5) creates a simple parametric passage from the class to the class .
The main goal of this paper is to study the Fekete-Szegö problem for the classes , . For the class , i.e., for , the Fekete-Szegö problem was examined in , where it was shown that
with sharpness of the result for . Recall here that in  Keogh and Merkes proved that
For Koepf in  extended the above result for the class of close-to-convex functions showing that
For the class and , we get the following sharp result published, among other results, in [, Theorem 2.3], namely
2 Main result
By we denote the class of all analytic functions p in of the form
having a positive real part in . For each , let
Inequalities (2.2) and (2.3) below are well known (see, e.g., [, pp.41, 166]).
Lemma 2.1 If is of the form (2.1), then
Both inequalities are sharp. The equality in (2.2) holds for every function , . The equality in (2.3) holds for every function
where and .
Now we prove the main theorem of this paper. The source of the method of proof is in Koepf’s paper , where the upper bound of for close-to-convex functions with λ restricted to the interval was calculated. However, we modify this technique and use it homogeneously for the class for all real λ, partially analogously as in  for the class , and in . We apply also the powerful Laguerre’s rule of counting zeros of polynomials in an interval. We propose Laguerre’s algorithm for such a computation by its simplicity, usefulness and efficiency.
consider a finite sequence , , of polynomials of the form
For each , let denote the number of sign changes in the sequence , . Given an interval , denote by the number of zeros of Q in I counted with their orders. Then the following theorem due to Laguerre holds.
Theorem 2.2 If , and , then or is an even positive integer.
Note that and . Thus, when , Theorem 2.2 reduces to the following useful corollary.
Corollary 2.3 If and , then or is an even positive integer, where and are the numbers of sign changes in the sequence of polynomial coefficients and in the sequence of sums , with , respectively.
The main theorem of the paper is as follows.
Theorem 2.4 Let . Then
For each and each , as well as for and each , the inequality is sharp and the equality is attained by a function in . In particular,
when , for each the second equality in (2.7) is attained by the function given by the differential equation(2.8)
when , for each the first equality in (2.7) is attained by the function , given by (2.8) with , i.e., when , by the function(2.9)
and when , by the Koebe function ;
when , for each the second equality in (2.7) is attained by the function(2.10)
and for each the first equality in (2.7) is attained by the function
Proof Fix . Observe from (1.5) that if and only if
for some and . Thus
Setting the series (1.1), (1.4) and (2.1) into (2.13), by comparing coefficients, we get
Let . Using (2.3), from (2.14) and (2.15), we have
Set and . Clearly, and, in view of (2.2), . It is convenient to use instead of λ in further computation. For , let
Set . For and , define
Consequently, in view of (2.16) we have
Now, for each and , we find the maximum value of on the rectangle R.
In the corners of R we have(2.17)(2.18)(2.19)
For and we have a linear function and for and we have a constant function.
For and , let
For we get the linear functions, so let . Then if and only if
Thus if and only if
Moreover, we have
For and , let
For we have the linear functions, evidently, so let . Note first that
Taking into account (2.22), we have
Using (2.23) we get
if and only if
i.e., in view of (2.22) if and only if
Since , so from the above we get the equation
Thus the solution of equation (2.25), and hence of (2.24), exists if and only if
Elementary computing shows that (2.26) holds if and only if
Thus the function has a critical point in , namely
as the unique solution of (2.25), if and only if (2.27) holds. Moreover,
We will prove that for each and the function has no critical point in .
Since and , we have
if and only if
Since , by comparing (2.29) and (2.22), we see that . By a simple observation, we deduce that the solution of (2.29) can exist only when
Squaring then (2.29), we obtain
Since by (2.22), for , taking into account (2.23), we have
Thus, by using (2.29) and (2.31), after simplifying we have
if and only if
It follows at once that for and , as well as for and , equation (2.32) has no root. Thus by (2.30) we consider
Solving now (2.32), we have
Clearly, and it remains to consider . It is easy to check that . Thus setting into (2.31) and computing, we have
A solution in of (2.34) exists if and only if
By (2.33) consider
We prove that then condition (2.35) is false. When , then an easy computation shows that the left-hand inequality in (2.35) is false. Thus by (2.36) it remains to consider
By an easy computation we check that then the left-hand inequality in (2.35) holds. Since , write the right-hand inequality in (2.35) as
The last step is to show, which does not cause difficulties, that under the assumption (2.37) the above inequality is false. We omit the details.
Summarizing, we proved that condition (2.35) is false, so equation (2.34) has no solution in .
Thus the proof that for and the function has no critical point in is finished.
Now we calculate the maximum value of in R, which, as was shown, is attained on the boundary of R. Let . Taking into account Part 3 with (2.20) and Part 4 with (2.27), we consider the following cases.
. Then the maximum value of is attained in a corner of R. Thus by (2.17)-(2.19) an easy computation shows that(2.38)
. Then the maximum value of is attained in a corner of R or in . Thus, by (2.17)-(2.19) and (2.21), we calculate that(2.39)
. Then the maximum value of is attained in a corner of R or in the point . Thus, by (2.17)-(2.19) and (2.28) with , we see that(2.40)
. Then we compare all values (2.17)-(2.19), and by (2.21) and (2.28), and . We will show that the value is the largest one. As it is easy to observe, it is enough to prove that(2.41)
i.e., in view of (2.21) and (2.28), after a simple computation, we have
As , so squaring (2.42) and computing, we equivalently have
To verify that (2.43) holds, setting , we will show that for every we have
() For , inequality (2.43) is evidently true.
() For , we have and inequality (2.44) after computing is equivalent to the evidently true inequality
() Let . We will show that for ,
Further, taking into account that and are continuous functions with
from (2.45) we deduce that
which confirms (2.44) and further (2.43).
Now we prove that (2.45) holds, i.e., after computation we have
As in (2.6), let , , be a sequence of polynomials of the form
corresponding to the polynomial in (2.5) for Laguerre’s rule in .
First we check the signs of the elements of the sequence , i.e., of the sequence for . A simple computation shows that for we have , , , , , , , , and . Hence(2.46)
Now we check the signs of the elements of the sequence for . After the detailed computation and arguments based on Laguerre’s rule, we show that , , , , , , , and . Moreover, we show that there exists a unique such that and for and for . Thus for three cases, namely for , and , we have
Hence, by (2.46) and by Corollary 2.3, we conclude that for each the polynomial has no zero in , and since , so (2.45) holds.
Now we shortly explain the method of describing the signs of , . Note that the case is evident since
For other cases, i.e., for , we use Laguerre’s rule in each case in the same manner. We clarify this for the case . We have
We will show that
i.e., after computing we get
To verify that (2.48) holds, we will show that
Applying Corollary 2.3, we see that
Indeed, the numbers of sign changes in the sequence of polynomial coefficients and in the sequence of sums , where , equal 3, i.e., . Thus we conclude that the polynomial w has no zero in the interval and, since , so (2.50) holds. Hence, and by the fact that , we deduce that (2.49) holds, which confirms (2.47).
Note here that for the case we show that the equation
has a unique solution . In this case, we show by using Laguerre’s rule that the corresponding polynomial w as in (2.50) has a unique zero in and further we deduce that
Summarizing, taking into account (2.38)-(2.41), we have
Finally, substituting , the above yields (2.7).
Now we deal with the sharpness of the result. Let . We prove that for each inequality (2.7) is sharp. Let . Since then
inequality (2.7) is of the form
Let . Then and, in view of (2.4), , with and . Setting and into (2.12), we get the function given by equation (2.8) for which, by (2.14) and (2.15),
which makes the equality in (2.51), so in (2.7). Clearly, because (2.12) is satisfied for .
Let . Then inequality (2.7) is of the form
Set and into (2.12). Then for we get the function given by (2.9) and for we get the Koebe function , with
which makes the equality in (2.52), so in (2.7). Clearly, for .
Let . We prove that for each inequality (2.7) is sharp. For inequality (2.7) is of the form
Setting and, by (2.4), into (2.12), we get the function given by (2.10) with and . Hence
which makes the equality in (2.53), so in (2.7).
For inequality (2.7) is of the form
Setting and into (2.12), we get the function given by (2.11) with and . Hence
which makes the equality in (2.54), so in (2.7). □
Remark 2.5 Particularly, from (2.7) for we have
For we have and , , and then Theorem 2.4 reduces to the result of  as follows.
For each , the inequality is sharp and the equality is attained by a function in . In particular, for each the second equality in (2.55) is attained by the function given by differential equation (2.8), where . For each , the first equality in (2.55) is attained by the Koebe function .
For we have and , and Theorem 2.4 yields the following.
For each the second equality in (2.56) is attained by the function given by (2.10). For each , the first equality in (2.56) is attained by function (2.11).
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The authors declare that they have no competing interests.
All authors read and approved the final manuscript.
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Kowalczyk, B., Lecko, A. The Fekete-Szegö inequality for close-to-convex functions with respect to a certain starlike function dependent on a real parameter. J Inequal Appl 2014, 65 (2014). https://doi.org/10.1186/1029-242X-2014-65
- Fekete-Szegö problem
- close-to-convex functions
- close-to-convex functions with respect to the Koebe function
- close-to-convex functions with argument δ
- functions convex in the positive direction of the imaginary axis