Some subclasses of close-to-convex functions

Theorem 2.1. If f ∈Cβ(α), α ∈[0, 1], β ∈(−π/2, π/2), then f is univalent in U. P r o o f. For α = 0 this is shown in [9] and [12]. Let now α ∈(0, 1]. The function ϕα(z) = 1 2α log 1 + αz 1 −αz , z ∈U, ϕα(0) = 0 , is convex and univalent in U. Moreover, if f ∈Cβ(α), where β ∈(−π/2, π/2), then Re  eiβ f ′(z) ϕ′α(z)

Theorem 2.2. If β ∈(−π/2, π/2), α1, α2 ∈[0, 1] and α1 ̸= α2, then Cβ(α1) ̸⊆Cβ(α2) and Cβ(α2) ̸⊆Cβ(α1). P r o o f. Let 0 ≤α2 < α1 ≤1. 1o. Let f be the solution of the equation (2.1) eiβ(1 −α2 1z2)f ′(z) = 1 + z2 1 −z2 cos β + i sin β, z ∈U , where β ∈(−π/2, π/2). Of course, f ∈Cβ(α1) and by (2.1) we have (2.2) Arg{eiβ(1−α2 2z2)f ′(z)} = Arg 1 + z2 1 −z2 cos β +i sin β

Theorem 2.3. If f ∈Cβ(α), α ∈(0, 1), β ∈(−π/2, π/2) and f is of the form (1.1), then, for all k ∈N, |a2k| ≤ 1 −α2k (1 −α2)k cos β , (2.11) |a2k+1| ≤2 cos β + (1 −2 cos β)α2k −α2(k+1) (1 −α2)(2k + 1) . (2.12) P r o o f. By (1.2) there exists a function q(z) = cos β + i sin β + ∞ X n=1

Corollary 2.4. If f ∈C(α), α ∈(0, 1), and f is of the form (1.1), then (2.19) |a2k| ≤ 1 −α2k (1 −α2)k and |a2k+1| ≤2 −α2k −α2(k+1) (1 −α2)(2k + 1) , k ∈N . The above results are sharp. The function fα,0(z) = 1 1 −α2  log 1 −α2z2

Corollary 2.5. If f ∈P ′(β), β ∈(−π/2, π/2), and f is of the form (1.1), then (2.20) |an| ≤2 n cos β, n ∈N . In particular, for β = 0, (2.21) |an| ≤2 n, n ∈N (see [7]). The estimates (2.20) and (2.21) can be obtained from (2.11) and (2.12) by putting α = 0. The following functions are extremal for the classes P ′(β) and P ′(0), respectively:

Corollary 2.6. If f ∈β-CV2(i), β ∈(−π/2, π/2), and f is of the form (1.1), then |a2k| ≤cos β , (2.22) |a2k+1| ≤2k cos β + 1 2k + 1 , k ∈N . (2.23) In particular, for β = 0, (2.24) |an| ≤1, n ∈N . The function f1,β(z) = lim

Theorem 3.1. If f ∈Cβ(α), α ∈[0, 1], β ∈(−π/2, π/2), then (3.1) |f ′(z)| ≤ p 1 + r4 + 2r2 cos 2β + 2r cos β (1 −α2r2)(1 −r2) = exp  ar sh 2r cos β 1 −r2  1 −α2r2 , (3.2)

Corollary 3.2. If f ∈C(α), α ∈(0, 1), then (3.8) 1 −r (1 + r)(1 + α2r2) ≤|f ′(z)| ≤ 1 + r (1 −r)(1 −α2r2) , (3.9) 1 1 + α2  log (1 + r)2 1 + α2r2 −(1 −α2) 1 α arctan(αr)  ≤|f(z)| ≤

Corollary 3.3. If f ∈P ′(0), then 1 −r 1 + r ≤|f ′(z)| ≤1 + r 1 −r , (3.10) 2 log(1 + r) −r ≤|f(z)| ≤−2 log(1 −r) −r (3.11) for z ∈U, |z| = r < 1. The functions h0,0(z) = −z −2 log(1 −z), z ∈U , and t0,0(z) = lim α→1 tα,0(z) = i log(1 −iz)2 −z, z ∈U ,

Corollary 3.4. If f ∈C0(1), then 1 −r (1 + r)(1 + r2) ≤|f ′(z)| ≤ 1 (1 −r)2 , (3.12) 1 2 log (1 + r)2 1 + r2 ≤|f(z)| ≤ r 1 −r (3.13) for z ∈U, |z| = r < 1. The functions

Corollary 3.5. If f ∈C(α), α ∈(0, 1], then f(U) contains the disk (3.14) |w| < 1 1 + α2  log 4 1 + α2 −(1 −α2) 1 α arctan α  (see [6]). The constant on the right hand side of (3.14) is best possible and the function tα,0 is extremal. For the class C(0) the following result is known (see [2]):

Corollary 3.6. If f ∈C(0), then f(U) contains the disk |w| < 2 log 2 −1 .

Corollary 3.7. If f ∈C(1), then f(U) contains the disk |w| < 1 2 log 2 . References [1] H. S. Al-Amiri and M. O. Reade, On a linear combination of some expressions in the theory of univalent functions, Monatsh. Math. 80 (4) (1975), 257–264. [2] I. M. Gal’perin, The theory of univalent functions with bounded rotation, Izv. Vyssh. Ucheb. Zaved. Mat. 1958 (3) (4), 50–61 (in Russian). [3] A. W. Goodman and E. B. Saff, On the definition of a close-to-convex function, Internat. J. Math. and Math. Sci.

Definition 1.1. A function f of the form (1.1) f(z) = z + a2z2 + . . . + anzn + . . . , z ∈U , regular in U belongs to the class Cβ(α), α ∈C, β ∈(−π/2, π/2), if (1.2) Re{eiβ(1 −α2z2)f ′(z)} > 0, z ∈U . We also set C(α) =