Abstract
For $0<p<\infty$ and $-2\leα\le0$ we show that the $L^p$ integral mean on $rD$ of analytic function in the unit disk $D$ with respect to the weighted area measure $(1-|z|^2)^αdA(z)$ is a logarithmically convex function of $r$ on $(0,1)$.
Results & Lemmas (7)
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Theorem 1.
Theorem 1. Suppose 0 < p < ∞, −2 ≤α ≤0, and f is analytic in D. Then the function Mp,α(f, r) is logarithmically convex. We have been unable…
Theorem 1. Suppose 0 < p < ∞, −2 ≤α ≤0, and f is analytic in D. Then the function Mp,α(f, r) is logarithmically convex. We have been unable to determine whether or not the range α ∈[−2, 0] is best possible. In other words, we do not know if there exists a set Ω properly containing [−2, 0] such that Mp,α(f, r) is logarithmically convex on (0, 1) for all p ∈(0, ∞), all α ∈Ω, and all f ∈H(D). It is certainly reasonable to expect that the logarithmic convexity of Mp,α(f, r) for all f will depend on
Lemma 2.
Lemma 2. Suppose f is positive and twice differentiable on (0, 1). Then (i) f(x) is convex in log x if and only if f ′(x) + xf ′′(x) ≥0
Lemma 2. Suppose f is positive and twice differentiable on (0, 1). Then (i) f(x) is convex in log x if and only if f ′(x) + xf ′′(x) ≥0
Lemma 3.
Lemma 3. Suppose f = f1/f2 is a quotient of two positive and twice dif- ferentiable functions on (0, 1). Then D(f(x)) = D(f1(x)) −D(f2(x))…
Lemma 3. Suppose f = f1/f2 is a quotient of two positive and twice dif- ferentiable functions on (0, 1). Then D(f(x)) = D(f1(x)) −D(f2(x)) (1) for x ∈(0, 1). Consequently, log f(x) is convex in log x if and only if D(f1(x)) −D(f2(x)) ≥0 (2) on (0, 1).
Lemma 2
Lemma 2, log f(x) is convex in log x if and only if inequality (2) holds. □
Lemma 2, log f(x) is convex in log x if and only if inequality (2) holds. □
Lemma 4.
Lemma 4. Suppose −2 ≤α ≤0 and x ∈[0, 1). Then (a) 1 −(α + 1)ϕ(x) −(1 −x)ϕ′(x) = 0. (b) ϕ(x) −x ≥0. (c) g1(x) ≡x(1 −x −αx) −(1 −x)ϕ(x) ≥0.…
Lemma 4. Suppose −2 ≤α ≤0 and x ∈[0, 1). Then (a) 1 −(α + 1)ϕ(x) −(1 −x)ϕ′(x) = 0. (b) ϕ(x) −x ≥0. (c) g1(x) ≡x(1 −x −αx) −(1 −x)ϕ(x) ≥0. (d) g2(x) ≡(α + 2)ϕ2(x) −2(1 + x + αx)ϕ(x) + 2x ≥0. (e) g3(x) ≡ϕ2(x) −(1 + x + αx)ϕ(x) + x ≥0.
Lemma 5.
Lemma 5. We have B2 −4AC > 0 on (0, 1).
Lemma 5. We have B2 −4AC > 0 on (0, 1).
Theorem 6.
Theorem 6. Let 0 < p < ∞and −2 ≤α ≤0. If M(x) is non-decreasing and log M(x) is convex in log x for x ∈(0, 1), then the function x 7→log Z…
Theorem 6. Let 0 < p < ∞and −2 ≤α ≤0. If M(x) is non-decreasing and log M(x) is convex in log x for x ∈(0, 1), then the function x 7→log Z x 0 M(t)(1 −t)αdt Z x 0 (1 −t)αdt is also convex in log x for x ∈(0, 1). The logarithmic convexity of Mp,α(f, r) is equivalent to following: if 0 < r1 < r2 < 1, 0 < θ < 1, and r = rθ 1r1−θ 2 , then
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