On sharp solutions to majorization and Fekete-Szego problems for starlike functions

Lemma 1 ([10]). The analytic function φ(z) for each |z| = r < 1 satisfies the fol- lowing inequality |φ′(z)| ≤1−|φ(z)|2 1−|z|2 = 1−|φ(z)|2 1−r2 (z ∈E). For some recent applications of this lemma, see [24,25].

Lemma 2 ([3]). Let w(z) = w1z+w2z2 +w3z3 +··· (z ∈E)

Lemma 3 ([3]). Let w(z) = w1z+w2z2 +w3z3 +··· (z ∈E) be a Schwarz function. Then for any complex number β, w2 −βw2 1 ≤max{1,|β|}. This estimate is sharp and attains equality by w(z) = z or w(z) = z2. 3. MAIN RESULTS In this section, we investigate the majorization problems for functions in the classes ST s(α) and ST c(α). The Fekete-Szeg˝o type problems are also solved for each func- tion in ST s(α) and ST c(α). Throughout the discussion, it is assumed that 0 ≤α < 1. 3.1. Majorization problems W

Theorem 1. Let f ∈A and g ∈ST s(α). If f is majorized by g for each z ∈E, then for |z| = r ≤rs, |f ′(z)| ≤|g′(z)|, where rs is the least positive root of the following equation [1−(1−α)sinr](1−r2)−2r = 0. (3.1)

Theorem 2. Let f ∈A and g ∈ST c(α). If f is majorized by g for each z ∈E, then for |z| = r ≤rc, |f ′(z)| ≤|g′(z)|, where rc is the least positive root of the following equation [1−(1−α)rcosr](1−r2)−2r = 0. (3.7)

Theorem 3. Let f ∈ST s(α) and 0 ≤α < 1. Then for any real number β, a3 −βa2 2 ≤    α−1 2
σ (β < σ1), 1−α 2 (σ1 ≤β ≤σ2),

Theorem 4. Let f ∈ST s(α) and 0 ≤α < 1. Then for any complex number β, a3 −βa2 2 ≤1−α 2 max{1,|σ|}, (3.15) where σ is given by (3.12).

Theorem 5. Let f ∈ST c(α) and 0 ≤α < 1. Then for any real number β, a3 −βa2 2 ≤    α−1 2 σ (β < σ1), 1−α 2 (σ1 ≤β ≤σ2), 1−α 2 σ

Theorem 6. Let f ∈ST c(α) and 0 ≤α < 1. Then for any complex number β, a3 −βa2 2 ≤1−α 2 max{1,|σ|}, where σ is given by (3.12).

Corollary 1. Let f ∈A and g ∈ST s(0) ≡S∗ s. If f is majorized by g for each z ∈E, then for |z| = r ≤rs, |f ′(z)| ≤|g′(z)| where rs is the least positive root of the following equation (1−r2)(1−sinr)−2r = 0.

Corollary 2. Let f ∈A and g ∈ST c(0) ≡S∗ c. If f is majorized by g for each z ∈E, then for |z| = r ≤rc, |f ′(z)| ≤|g′(z)| where rc is the least positive root of the following equation 1−r2
(1−rcosr)−2r = 0.

Corollary 3. Let f ∈ST s(0) ≡S∗ s or f ∈ST c(0) ≡S∗ c. Then for any real number β, a3 −βa2 2 ≤    −2β−1 2 (β < 0), 1 2

Corollary 4. Let f ∈ST s(0) ≡S∗ s or f ∈ST c(0 ≡S∗ c. Then for any complex number β, a3 −βa2 2 ≤max 1 2, |2β−1| 2  .