On the third logarithmic coefficient in some subclasses of close-to-convex functions

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(1) and S be the subclass of A of all univalent functions.
The numbers γ n are called logarithmic coefficients of f . As is well known, the logarithmic coefficients play a crucial role in Milin conjecture ( [23], see also [10, p. 155]), namely that for f ∈ S, n m=1 n k=1 k|γ k | 2 − 1 k ≤ 0.
De Branges [8] showing Milin conjecture confirmed the famous Bieberbach conjecture (e.g., [10, p. 37]). It is surprising that for the class S the sharp estimates of single logarithmic coefficients S are known only for γ 1 and γ 2 , namely, and are unknown for n ≥ 3.
As usual, instead of the whole class S one can take into account their subclasses for which the problem of finding sharp estimates of logarithmic coefficients can be studied. When f ∈ S * , the class of starlike functions, the inequality |γ n | ≤ 1/n holds for n ∈ N (see e.g. [30, p. 42]). Moreover, for f ∈ SS * (β), the class of strongly starlike function of order β (0 < β ≤ 1), it holds that |γ n | ≤ β/n (n ∈ N) (see [28]). Also, the bounds of γ n for functions in the class of gamma-starlike functions, close-to-convex functions and Bazilevič functions were examined in [30, p. 116], [9,27,29], respectively. In two recent papers, namely, in [15] the bounds of early logarithmic coefficients of the subclasses F 1 , F 2 , F 3 of S and in [1] of the subclass F 4 of S of functions f satisfying respectively the condition Re were computed. Let us note that each class defined above is the subclass of the well known class of close-to-convex functions, so therefore families F i , i = 1, . . . , 4, contain only univalent functions (e.g., [12, Vol. II, p. 2]). Both cited paper contains sharp bounds of γ 1 and γ 2 and partial results for γ 3 only. The first three results in theorem below were shown in [15], and the last one in [1]. 3. if f ∈ F 3 and 1/2 ≤ a 2 ≤ 3/2, then 4. if f ∈ F 4 and 1 ≤ a 2 ≤ 2, then In this paper we improve all results in Theorem 1 for γ 3 for the general case when a 2 is real. Differentiating (2) and using (1) we get Since each class F i , i = 1, . . . , 4, has a representation by using the Carathéodory class P, i.e., the class of functions p ∈ H of the form having a positive real part in D, the coefficients of functions in F i , so γ 3 has a suitable representation expressed by the coefficients of functions in P. Therefore to get the upper bound of γ 3 our computing is based on parametric formulas for the second and third coefficients in P. The proof of results of Theorem 1 are based on the well known formula on c 2 and on the formula c 3 due to Libera and Zlotkiewicz [21,22] with the restriction that c 1 ≥ 0. Since all classes F i are not rotation invariant, to omit the assumption c 1 ≥ 0. we will use a general formula for c 3 , which was found in [4]. However to be self contained we present a proof for c 3 here. Moreover in our computation of the sharp bound of γ 3 we use a lemma due to Ohno and Sugawa [24]. Let us mention that the conditions (3), (4) and (6) were discovered by Ozaki [25] as useful criteria of univalence. Recall also that the classes F 2 and F 4 have nice geometrical interpretations, and therefore they play an important role in the geometric function theory. Each function f ∈ F 2 maps univalently D onto a domain f (D) convex in the direction of the imaginary axis, i.e., for every w 1 , w 2 ∈ f (D) such that Re w 1 = Re w 2 the line segment [w 1 , w 2 ] lies in f (D), with the additional property that there exist two points ω 1 , [12, p. 199]). Each function in the class F 4 maps univalently D onto a domain f (D) called convex in the positive direction of the real axis, i.e., {w + it : t ≥ 0} ⊂ f (D) for every w ∈ f (D) [2,6,7,11,18,19].
At the end, let us say that the conditions (3)-(6) were generalized by replacing polynomials standing at f by any quadratic polynomial [16,17], and by any polynomial of any degree having their roots in C\D [13,14].

Lemma 1 If p ∈ P is of the form
and For ζ 1 ∈ T, there is a unique function p ∈ P with c 1 as in (9), namely, For ζ 1 ∈ D and ζ 2 ∈ T, there is a unique function p ∈ P with c 1 and c 2 as in (9)-(10), namely, For ζ 1 , ζ 2 ∈ D and ζ 3 ∈ T, there is a unique function p ∈ P with c 1 , c 2 and c 3 as in (9)-(11), namely, The next lemma is a special case of more general results due to Choi, Kim and Sugawa [5] (see also [24]). Define

Logarithmic coefficients
Now we will prove the main results of this paper.

The class F
Theorem 3 If f ∈ F 2 is of the form (1) with a 2 ∈ R, then The inequality is sharp with the extremal function where Proof Let f ∈ F 2 be of the form (1). Then there exists p ∈ P of the form (8) such that Substituting the series (1) and (8) into (35) by equating the coefficients we get Note first that since a 2 is real, so is c 1 and (9) holds with some ζ 1 ∈ [−1, 1]. Moreover, from (36) it follows that a 2 ∈ [−1, 1]. By (7) and (36) we get Using now (9)-(11) we have with ζ 1 ∈ [−1, 1] and ζ 2 , ζ 3 ∈ D.
Hence for ζ 1 = 1, ζ 1 = −1 and ζ 1 = 0 we respectively have Now let ζ 1 ∈ (−1, 1)\{0} =: I . Then from (37) we obtain where Ψ is defined by (23) with Note now that AC < 0 for ζ 1 ∈ I . Moreover, Therefore by Lemma 2 we get Hence and from (39) it follows that where We note that the function ϕ is even in I. As easy to verify This and (38) show that the inequality (32) is true. By tracking back the above proof, we see that equality in (32) holds when it is satisfied that and |A + Bζ 2 + Cζ 2 2 | + 1 − |ζ 2 | 2 = 1 + |A| + where Indeed we can easily check that one of the solutions of the equation (42) is By Lemma 1 a function p of the form (14) with ζ i (i ∈ {1, 2, 3}) given by (41) and (43), i.e., the function (34) belongs to P. Thus the function (33) belongs to F 2 . Substituting (41) and (43) into (37) we get equality in (32). This ends the proof of the theorem.

The class F 3
Recall that f ∈ F 3 if f ∈ A and Theorem 4 If f ∈ F 3 is of the form (1) with a 2 ∈ R, then This result is sharp.
Summarizing, (63), (65) and (67) show that the inequality (57) is true. A simlar method used for the proof of Theorem 2, the equality in (57) when it is satisfied that By Lemma 1 a function p of the form (14) with ζ i (i ∈ {1, 2, 3}) given by (68), i.e., the function (59) belongs to P. Thus the function (58) belongs to F 4 . Substituting (68) into (62) we get equality in (57). This ends the proof of the theorem.