Acta Univ. Sapientiae, Mathematica, 11, 1 (2019) 203–223

Lemma 1 (see [21]) If a > b > r ≥|z| , and λ ∈[0, 1], then

Theorem 1 Let β, ρ > 0, α ∈[0, 1) and k ≥0. Then, the following state- ments are valid: a. The radius of k-uniform convexity of order α of the function fρ,β is the real number ruc k,α(fρ,β) which is the smallest positive root of the equation (1 + k)r Ψ′′ ρ,β(r) Ψ′ ρ,β + ( 1 β −1)(1 + k)r Ψ′ ρ,β(r) Ψρ,β(r) + 1 −α = 0

Theorem 5] given by Γ(β)Ψρ,β(z) = zβ Y n≥1

Theorem 2 Let ν > −1, s ∈{2, 3} and q ∈(0, 1). Then, the following asser- tions holds true a. Suppose that ν > 0. Then, the radius of k−uniform convexity of order α of the function z 7→f(s) ν (z; q) is the real number ruc k,α(f(s) ν ) which is the smallest positive root of the equation 1 −α + (1 + k)r(f(s) ν (r; q))′′ (f(s) ν (r; q))′ = 0 in (0, j′ ν,1(q)). b. The radius of k-uniform convexity of order α of the function z 7→g(s)

Lemma 2 [19] If a is any point in |arg w| ≤πγ 2 and if Ra ≤Re[a] sin πγ 2 −Im[a] cos πγ 2 , Im[a] ≥0, the disk |w −a| ≤Ra is contained in the sector |arg w| ≤πγ 2 , 0 < γ ≤1. In particular when Im[a] = 0, the condition becomes Ra ≤a sin πγ 2 .

Theorem 3 Let ρ > 0 and β > 0. The following assertions are true: a. The radius of strong starlikeness of fρ,β is the smallest positive root of the equation 2 β X n≥1 r2  λ2 ρ,β,n + r2 sin πγ 2  λ4 ρ,β,n −r4 −sin πγ

Theorem 4 Let ν > −1, s ∈{2, 3} and q ∈(0, 1). Moreover, let ην,n(q) be the nth positive root of the function z 7→J(s) ν (z; q). Then the following assertions are true: a. The radius of strong starlikeness of the function f(s) ν (z; q) is the smallest positive root of the equation 2 ν X n≥1 r2 η2 ν,n(q) + r2 sin πγ 2