The Fekete-Szegö inequality for close-to-convex functions with respect to a certain starlike function dependent on a real parameter
© Kowalczyk and Lecko; licensee Springer. 2014
Received: 9 December 2013
Accepted: 31 January 2014
Published: 13 February 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 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]).
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.
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], ).
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 .
2 Main result
Inequalities (2.2) and (2.3) below are well known (see, e.g., [, pp.41, 166]).
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.
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.
- (i)when , for each the second equality in (2.7) is attained by the function given by the differential equation(2.8)
- (ii)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)
- (iii)when , for each the second equality in (2.7) is attained by the function(2.10)
- 1.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.
- 3.For and , let
- 4.For and , let
We will prove that for each and the function has no critical point in .
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 .
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.
- (A). 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)
- (B). 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)
- (C). 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)
- (D). 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)
() For , inequality (2.43) is evidently true.
which confirms (2.44) and further (2.43).
- (a)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)
- (b)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.
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).
Finally, substituting , the above yields (2.7).
which makes the equality in (2.51), so in (2.7). Clearly, because (2.12) is satisfied for .
which makes the equality in (2.52), so in (2.7). Clearly, for .
which makes the equality in (2.53), so in (2.7).
which makes the equality in (2.54), so in (2.7). □
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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