Abstract
Let $\es$ be the class of analytic and univalent functions in the unit disk $|z|<1$, that have a series of the form $f(z)=z+ \sum_{n=2}^{\infty}a_nz^n$. Let $F$ be the inverse of the function $f\in\es$ with the series expansion %in a disk of radius at least $1/4$ $F(w)=f^{-1}(w)=w+ \sum_{n=2}^{\infty}A_nw^n$ for $|w|<1/4$. The logarithmic inverse coefficients $Γ_n$ of $F$ are defined by the formula $\log\left(F(w)/w\right)\,=\,2\sum_{n=1}^{\infty}Γ_n(F)w^n$. % In this paper, we determine the log
Results & Lemmas (18)
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Theorem 1.
Theorem 1. Let f ∈S (or S∗) and F be the inverse function of f and have the form (1.2). Then for n ∈N, the logarithmic inverse coefficients…
Theorem 1. Let f ∈S (or S∗) and F be the inverse function of f and have the form (1.2). Then for n ∈N, the logarithmic inverse coefficients Γn of F satisfy the sharp inequality |Γn(F)| ≤1 2n 2n n . Equality is attained if and only if f is the Koebe function or one of its rotations.
Theorem 2.
Theorem 2. Let f ∈S∗(A, B), δ = (1 −A)/(1 −B) with −1 ≤B < A ≤1, and kA,B;n(z) = z(1 + Bzn)(A−B)/nB. Then for n ∈N, the logarithmic inverse…
Theorem 2. Let f ∈S∗(A, B), δ = (1 −A)/(1 −B) with −1 ≤B < A ≤1, and kA,B;n(z) = z(1 + Bzn)(A−B)/nB. Then for n ∈N, the logarithmic inverse coefficients Γn of F satisfy the following inequalities:
Theorem 2
Theorem 2 for the case A = 1 −2β and B = −1 takes the following simple form.
Theorem 2 for the case A = 1 −2β and B = −1 takes the following simple form.
Corollary 3.
Corollary 3. Let f ∈S∗(β) for some β ∈[0, 1), and kβ;n(z) = z/(1 −zn)2(1−β)/n. Then the logarithmic inverse coefficients Γn of F satisfy the…
Corollary 3. Let f ∈S∗(β) for some β ∈[0, 1), and kβ;n(z) = z/(1 −zn)2(1−β)/n. Then the logarithmic inverse coefficients Γn of F satisfy the inequalities: (1) for n ∈N and β ∈[0, 1/n), we have (3.4) |Γn(F)| ≤1 2n n−1 Y j=0 2n(1 −β) −j 1 + j , (2) for n ∈N and β ∈Ik(n), k = 1, 2, . . . , n −1, we have |Γn(F)| ≤n −k 2n2
Theorem 2
Theorem 2 to prove the following.
Theorem 2 to prove the following.
Theorem 4.
Theorem 4. Let f ∈Sα(β) for some β ∈[0, 1) and α ∈(−π/2, π/2). Then the logarithmic inverse coefficients of F satisfy the inequalities: (1)…
Theorem 4. Let f ∈Sα(β) for some β ∈[0, 1) and α ∈(−π/2, π/2). Then the logarithmic inverse coefficients of F satisfy the inequalities: (1) for n ∈N and β ∈I0(n) = [0, 1/n), we have (3.7) |Γn(F)| ≤1 2n n−1 Y j=0 |2n(1 −β)e−iα cos α −j| 1 + j (2) for n ∈N and β ∈Ik(n), k = 1, 2, . . . , n −1, we have (3.8) |Γn(F)| ≤n −k 2n2
Lemma 5.
Lemma 5. Let f ∈G(c) for some c ∈(0, 1] and for each fixed λ > 0, the Taylor coefficients bm(λ, f) be given by (2.1). Then (1) for λ ∈(0, 1],…
Lemma 5. Let f ∈G(c) for some c ∈(0, 1] and for each fixed λ > 0, the Taylor coefficients bm(λ, f) be given by (2.1). Then (1) for λ ∈(0, 1], we have (3.10) |bm(λ, f)| ≤ λc m(1 + c) for m = 1, 2, . . . ; (2) for λ > 1, we have (3.11) |bm(λ, f)| ≤ 1 (1 + c)m m−1 Y
Corollary 6.
Corollary 6. Let f ∈G(1) and for each fixed λ > 0, let the Taylor coefficients bm(λ, f) be given by (2.1). Then (1) for λ ∈(0, 1], we have…
Corollary 6. Let f ∈G(1) and for each fixed λ > 0, let the Taylor coefficients bm(λ, f) be given by (2.1). Then (1) for λ ∈(0, 1], we have (3.17) |bm(λ, f)| ≤λ 2m for m = 1, 2, . . . ; (2) for λ > 1, we have (3.18) |bm(λ, f)| ≤1 2m m−1 Y j=0 λ + j
Theorem 7.
Theorem 7. Let f ∈G(c) for some c ∈(0, 1]. Then the logarithmic inverse coefficients Γn of F satisfy the inequality |Γn(F)| ≤ 1 2n(1 + c)n…
Theorem 7. Let f ∈G(c) for some c ∈(0, 1]. Then the logarithmic inverse coefficients Γn of F satisfy the inequality |Γn(F)| ≤ 1 2n(1 + c)n n−1 Y j=0 nc + j (1 + j) for n ∈N. The result is best possible for the function f ′ c(z) = (1 −z)c.
Lemma 5.
Lemma 5. For λ = n ∈N, we note that the inequalities (3.10) and (3.11) are applicable. Therefore, the inequalities (3.10) and (3.11) yield…
Lemma 5. For λ = n ∈N, we note that the inequalities (3.10) and (3.11) are applicable. Therefore, the inequalities (3.10) and (3.11) yield |Γn(F)| = 1 2n|bn(n, f)| for n ∈N.
Corollary 8.
Corollary 8. Let f ∈G(1). Then |Γn(F)| ≤(2n −1)! (n!)2 2n+1 for n ∈N. The result is best possible for the function f0(z) = z −z2/2. 3.5.…
Corollary 8. Let f ∈G(1). Then |Γn(F)| ≤(2n −1)! (n!)2 2n+1 for n ∈N. The result is best possible for the function f0(z) = z −z2/2. 3.5. Logarithmic inverse coefficients for U(λ). Now, we will discuss the logarithmic inverse coefficients Γn for the class U(λ). It is a simple exercise to see that f ∈U(λ) if and only if (3.19) f(z) = z 1 −a2z + λz R z 0 ω(t) dt, where 2a2 = f ′′(0), ω is analytic and |ω(z)| ≤1 for |z| < 1. Moreover, we also see from
Theorem 9.
Theorem 9. Let f ∈U(λ) for 0 < λ ≤1. Then the logarithmic inverse coefficients Γn of F satisfy the inequality |Γ1(F)| ≤1 2 [1 + λv(|a|)] and…
Theorem 9. Let f ∈U(λ) for 0 < λ ≤1. Then the logarithmic inverse coefficients Γn of F satisfy the inequality |Γ1(F)| ≤1 2 [1 + λv(|a|)] and |Γ2(F)| ≤1 4 (1 + λv(|a|))2 + 2λ|a| . Equality is achieved in both inequalities for the function (3.21) f(z) = z 1 −(1 + λv(a))z + λz R z
Theorem 10.
Theorem 10. Let f ∈F(α) for some α ∈[−1/2, 1). Then |Γ1(F)| ≤1 −α 2. Equality is attained if and only if f ′(z) = (1 −z)−2(1−α) or a…
Theorem 10. Let f ∈F(α) for some α ∈[−1/2, 1). Then |Γ1(F)| ≤1 −α 2 . Equality is attained if and only if f ′(z) = (1 −z)−2(1−α) or a rotation of this function. The second and the third relations in (3.26) give
Theorem 11.
Theorem 11. Let f ∈F(α) for some α ∈[−1/2, 1). Then (a) If α ∈[−1/2, 1/5], then |Γ2(F)| ≤(1 −α)(3 −5α) 12. Equality is attained in each…
Theorem 11. Let f ∈F(α) for some α ∈[−1/2, 1). Then (a) If α ∈[−1/2, 1/5], then |Γ2(F)| ≤(1 −α)(3 −5α) 12 . Equality is attained in each case if and only if f ′(z) = (1 −z)−2(1−α) or a rotation of this function. (b) If α ∈(1/5, 1), then |Γ2(F)| ≤1 −α 6 . Equality is attained in each case if and only if f ′(z) = (1 −z2)−(1−α) or a rotation of this function.
Lemma 12.
Lemma 12. [23, Lemma 2] Let ϕ(z) = P∞ k=1 ckzk ∈B be a Schwarz function and Ψ(ϕ) = |c3 + µc1c2 + υc3 1|. Then we have the following sharp…
Lemma 12. [23, Lemma 2] Let ϕ(z) = P∞ k=1 ckzk ∈B be a Schwarz function and Ψ(ϕ) = |c3 + µc1c2 + υc3 1|. Then we have the following sharp estimates: (a) Ψ(ϕ) ≤1 if (µ, υ) ∈D1 ∪D2, where D1 = (µ, υ) ∈R2 : |µ| ≤1 2, −1 ≤υ ≤1 , and D2 =
Theorem 13.
Theorem 13. Let f ∈F(α) for α ∈[−1 2, 1). Then |Γ3(F)| ≤1 −α 12, α ∈ 0.21605468, 7 10 . Equality is attained if f ′(z) = (1 −z3)−2(1−α)…
Theorem 13. Let f ∈F(α) for α ∈[−1 2, 1). Then |Γ3(F)| ≤1 −α 12 , α ∈ 0.21605468, 7 10 . Equality is attained if f ′(z) = (1 −z3)−2(1−α) 3 or a rotation of this function. Also, |Γ3(F)| ≤(1 −α)(3α −2)(2α −1) 12
Corollary 14.
Corollary 14. Let f ∈C. Then |Γn(F)| ≤ 1 2n for n = 1, 2, 3. The estimates are sharp for the function l(z) = z/(1 −z).
Corollary 14. Let f ∈C. Then |Γn(F)| ≤ 1 2n for n = 1, 2, 3. The estimates are sharp for the function l(z) = z/(1 −z).
Corollary 15.
Corollary 15. If f ∈F(−1/2), then we have the sharp inequalities |Γ1(F)| ≤3 4, |Γ2(F)| ≤11 16, and |Γ3(F)| ≤7 8 The estimates are sharp for…
Corollary 15. If f ∈F(−1/2), then we have the sharp inequalities |Γ1(F)| ≤3 4, |Γ2(F)| ≤11 16, and |Γ3(F)| ≤7 8 The estimates are sharp for the function f0(z) = z −z2/2 (1 −z)2 . 4. Concluding Remarks From Theorem 9, we see that logarithmic inverse coefficients for the family U(λ) for the remaining coefficients Γn for n ≥3 are open. We recognized that in the case of convex functions f ∈C, |Γn(F)| ≤ 1 2n cannot be valid for n ≥10, although this is true for n = 1, 2, 3 by Corollary 14. In fact, if
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