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Abstract
In this paper we study class $\mathcal{S}^+$ of univalent functions $f$ such that $\frac{z}{f(z)}$ has real and positive coefficients. For such functions we give estimates of the Fekete-Szegő functional and sharp estimates of their initial coefficients and logarithmic coefficients. Also, we present necessary and sufficient conditions for $f\in \mathcal{S}^+$ to be starlike of order $1/2$.
Results & Lemmas (7)
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Theorem 1.
Theorem 1. For each f ∈S+ we have −1 ≤a3 −γa2 2 ≤ ( 1 + 2e−2γ/(1−γ), 0 ≤γ ≤ ν0 1+ν0 = 0.456278... 2(1 −γ) (ν0+1)2 2ν0+1, ν0 1+ν0 ≤γ < 1.…
Theorem 1. For each f ∈S+ we have −1 ≤a3 −γa2 2 ≤ ( 1 + 2e−2γ/(1−γ), 0 ≤γ ≤ ν0 1+ν0 = 0.456278 . . . 2(1 −γ) (ν0+1)2 2ν0+1 , ν0 1+ν0 ≤γ < 1. where ν0 = 0.83927 . . . is the positive real root of the equation (6) 2(2ν + 1)e−2ν = 1.
Theorem 2.
Theorem 2. Let f ∈S+ and let γ1, γ2, γ3 be its logarithmic coefficients. Then (a) −1 ≤γ1 ≤0; (b) −1 2 ≤γ2 ≤ (ν0+1)2 2(2ν0+1) = 0.631464...,…
Theorem 2. Let f ∈S+ and let γ1, γ2, γ3 be its logarithmic coefficients. Then (a) −1 ≤γ1 ≤0; (b) −1 2 ≤γ2 ≤ (ν0+1)2 2(2ν0+1) = 0.631464 . . ., where ν0 = 0.83927 . . . is the solution of the equation (6); (c) −1 4 ≤γ3 ≤1 3. Some of these results are the best possible.
Theorem 3.
Theorem 3. If f(z) = z + a2z2 + a3z3 + · · · belongs to the class U+(λ), 0 < λ ≤1, then we have −(1 + λ) ≤a2 ≤0, −λ ≤a3 ≤1 + λ + λ2, −(1 +…
Theorem 3. If f(z) = z + a2z2 + a3z3 + · · · belongs to the class U+(λ), 0 < λ ≤1, then we have −(1 + λ) ≤a2 ≤0, −λ ≤a3 ≤1 + λ + λ2, −(1 + λ + λ2 + λ4) ≤a4 ≤4λ 3 r 2λ 3 , a5 ≥ −λ/3, 0 < λ ≤4/27 −9λ2/4, 4/27 ≤λ ≤1 . All these inequalities are sharp.
Corollary 1.
Corollary 1. Let f(z) = z + a2z2 + a3z3 + · · · belong to the class S+. Then we have the next sharp inequalities −2 ≤a2 ≤0, −1 ≤a3 ≤3, −4…
Corollary 1. Let f(z) = z + a2z2 + a3z3 + · · · belong to the class S+. Then we have the next sharp inequalities −2 ≤a2 ≤0, −1 ≤a3 ≤3, −4 ≤a4 ≤4 3 r 2 3, −9 4 ≤a5 ≤5. We note that upper bound for a5 follows from de Brange’s theorem. 3. Relation with starlike functions In this section we study the relation between the class S+ and the class of starlike functions.
Theorem 4.
Theorem 4. Let f ∈A and satisfy the condition (1). Then the condition (11) ∞ X n=1 (2n −1)bn ≤1 is necessary and sufficient for f to be in…
Theorem 4. Let f ∈A and satisfy the condition (1). Then the condition (11) ∞ X n=1 (2n −1)bn ≤1 is necessary and sufficient for f to be in the class S⋆(1/2).
Theorem 5.
Theorem 5. Let f ∈S+ and let b1 = 0, then f ∈S⋆.
Theorem 5. Let f ∈S+ and let b1 = 0, then f ∈S⋆.
Theorem 6.
Theorem 6. Let f ∈S+. Then the function (12) g(z) = z + 1 2 z f(z) −1 −b1z is univalent in D. More precisely, Re g′(z) > 0 (z ∈D), g…
Theorem 6. Let f ∈S+. Then the function (12) g(z) = z + 1 2 z f(z) −1 −b1z is univalent in D. More precisely, Re g′(z) > 0 (z ∈D), g ∈S⋆and g ∈U.
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