On the Fekete–Szegö type functionals for functions which are convex in the direction of the imaginary axis

Theorem 1. If f ∈C0(h), then f is convex in the direction of the imaginary axis.

Theorem 2. Let all coefficients of f given by (1) be real. Then, f ∈C0(h) ⇔f ∈KR(i). In the above, KR(i) denotes the class of functions of the form (1) which are convex in the direction of the imaginary axis and have all real coefficients. Robertson [26] proved that f ∈KR(i) ⇔z f ′(z) ∈T . (8)

Theorem 3. If f ∈C0(h) is of the form (1), then |an| ≤1.

Lemma 4 ([7]). If p ∈P and µ ∈R, then the following sharp estimates hold (1) |pn+m −µpmpn| ≤2 for n,m = 1,2, ..., (2) |p2 −µp12| ≤2.

Lemma 5 ([17]). If p ∈P , then (1) 2p2 = p12 + x(4−p12), (2) 4p3 = p13 +2p1(4−p12)x −p1(4−p12)x2 +2(4−p12)(1−|x|2)y, for some x and y such that |x| ≤1, |y| ≤1. Since ¯¯p2 2 −µp1p3 ¯¯ ≤ ¯¯p2 2 −p4 ¯¯+ ¯¯p4 −µp1p3 ¯¯ , directly from Lemma 4 we conclude the following fact.

Lemma 6. If p ∈P , then the sharp estimate |p22 −µp1p3| ≤4 holds for µ ∈[0,1]. Let us return to the correspondence between the functions in C0(h) and P . Rewriting it as follows ¡ 1−z2¢ f ′(z) = p(z), f ∈C0(h), p ∈P , (13) we can express Φf (µ) and Θf (µ) for f ∈C0(h) in terms of the coefficients of p ∈P : Φf (µ) = 1 8 p1 ¡ p1 + p3 ¢ −1

Theorem 7. If f ∈C0(h) is of the form (1) and a2 is a real number, then ¯¯Φf (µ) ¯¯ ≤ ( 1−µ, µ ≤1/2 µ, µ ≥1/2. Equality holds for the functions f (z) = z 1−z and f (z) = z 1+z if µ ≤1/2 and for f (z) = z 1−z2 if µ ≥1/2.

Corollary 8. If f ∈C0(h) is of the form (1) and a2 is a real number, then ¯¯a2a4 −a3 2¯¯ ≤1. Equality holds if f (z) = z 1−z2 . 4. Estimation of |Θf (µ)| At the beginning of this section, observe that ¯¯Θf (µ) ¯¯ ≤ ¯¯1−µ ¯¯ for µ ≤0 or

Lemma 9. If ac ≥0, then Y (a,b,c) = ( |a|+|b|+|c|, |b| ≥2(1−|c|) , 1+|a|+ b2 4(1−|c|) , |b| < 2(1−|c|). If ac < 0, then Y (a,b,c) =    

Lemma 10. The curve a = 0 in Ωcoincides with the image of [2/3,1) by the descreasing function p(γ) = 2 q 1−γ 2γ−1.

Lemma 11. The inequality b < 2(1+|c|) holds for all (γ,p) ∈Ω.

Lemma 12. The inequalities a > 0, b2 ≥−4a(1 −c2)/c, b ≥2(1 + c) and ab ≥−c(b + 4a) are contradictory in Ω. The first two lemmas are easy to verify.

Lemma 13. max © h3(γ,p) : (γ,p) ∈Ω4 ª = ( 1−3 2γ, γ ∈ ¡ 0, ¡p 29−5 ¢

Lemma 15. For (γ,p) ∈Ω1 ∪Ω6, we have h1(γ,p) < 1 2.

Lemma 16. For (γ,p) ∈Ω2 ∪Ω3, we have h2(γ,p) ≤1 2.

Lemma 17. For (γ,p) ∈Ω7, we have h4(γ,p) < 1 2.

Theorem 18. If f ∈C0(h) is of the form (1) and a2 is a real number, then ¯¯Θf (µ) ¯¯ ≤        1−µ, µ ≤µ0 F(µ), µ ∈ £

Corollary 19. If f ∈C0(h) is of the form (1) and a2 is a real number, then |a4 −a2a3| ≤125 243 . C. R. Mathématique, 2020, 358, n11-12, 1213-1226

Corollary 20. If f ∈KR(i) is of the form (1), then ¯¯Φf (µ) ¯¯ ≤ ( 1−µ, µ ≤1/2 µ, µ ≥1/2 .

Corollary 21. Let F and µ0 be defined as in Theorem 2. If f ∈KR(i) is of the form (1), then ¯¯Θf (µ) ¯¯ ≤        1−µ, µ ≤µ0 F(µ), µ ∈[µ0,3/2] µ−1,

Theorem 22. If f ∈Q(1/2) is of the form (1), then |Φf (µ)| ≤        1−µ, µ ≤0 1, µ ∈[0,1] µ, µ ≥1 ,

Theorem 23. If f ∈Q(1/2) is of the form (1), then |Φf (µ)| ≤µ for µ ∈[3/4,1]. This result is sharp. For µ ∈[(2 − p 2)/4,3/4] the bound is (9 −16µ + 8µ2)/8(1 −µ), but it is not sharp. What is interesting here, they conjectured that the exact bound of Φf (µ) is the same as in the assertion of Corollary 20. Unfortunately, this conjecture is false. It can be proved that, for example, if µ = 1/2 the sharp inequality |Φf (1/2)| ≤5/8 holds for Q(1/2), contrary to the conjectured value 1/2. This shows t

Corollary 24. If g ∈T is of the form (27), then ¯¯Φg (λ) ¯¯ ≤ ( 8−9λ, λ ≤4/9 9λ, λ ≥4/9. The bound is sharp. Equality holds for the functions g(z) = z (1−z)2 and g(z) = z (1+z)2 if λ ≤4/9 and for f (z) = z(1+z2) (1−z2)2 if λ ≥4/9.

Corollary 25. Let G(λ) = 2 27λ · 2 p 7−7λ+λ23 +(2λ−1) ¡ 34−10λ+λ2¢¸ (28) and λ0 = ( p 29−5)/2 = 0.192.... If g ∈T is of the form (27), then

Corollary 26. If g ∈T is of the form (27), then ¯¯b2b4 −b3 2¯¯ ≤9. Equality holds if g(z) = z(1+z2) (1−z2)2 .

Corollary 27. If g ∈T is of the form (27), then |b4 −b2b3| ≤2. Equality holds if g(z) = z (1−z)2 or g(z) = z (1+z)2 . The bound from Corollary 26 coincides with the result from [29] which was obtained in a quite different way. The result from Corollary 27 coincides with a particular case of the generalized Zalcman conjecture for the class T which was proved in [19] by Ma. C. R. Mathématique, 2020, 358, n11-12, 1213-1226