Abstract
We show that the $L^2$ integral mean on $r\D$ of an analytic function in the unit disk $\D$ with respect to the weighted area measure $(1-|z|^2)^α\,dA(z)$, where $-3\leα\le0$, is a logarithmically convex function of $r$ on $(0,1)$. We also show that the range $[-3,0]$ for $α$ is best possible.
Results & Lemmas (8)
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Lemma 1.
Lemma 1. Suppose f is twice differentiable on (0, 1). Then f(x) is convex in log x if and only if f ′(x) + xf ′′(x) ≥0 on (0, 1).
Lemma 1. Suppose f is twice differentiable on (0, 1). Then f(x) is convex in log x if and only if f ′(x) + xf ′′(x) ≥0 on (0, 1).
Corollary 2.
Corollary 2. Suppose f is twice differentiable on (0, 1). Then f(x) is con- vex in log x if and only if f(x2) is convex in log x.
Corollary 2. Suppose f is twice differentiable on (0, 1). Then f(x) is con- vex in log x if and only if f(x2) is convex in log x.
Corollary 3.
Corollary 3. Suppose f is positive and twice differentiable on (0, 1). Then the function log f(x) is convex in log x if and only if D(f(x))…
Corollary 3. Suppose f is positive and twice differentiable on (0, 1). Then the function log f(x) is convex in log x if and only if D(f(x)) =: f ′(x) f(x) + x f ′(x) f(x) ′ = f ′(x) f(x) + xf ′′(x) f(x) −x f ′(x) f(x) 2 is nonnegative on (0, 1).
Proposition 4.
Proposition 4. Suppose k ≥0, −2 ≤α ≤0, and 0 < p < ∞. Then the function log Mp,α(zk, r) is convex in log r.
Proposition 4. Suppose k ≥0, −2 ≤α ≤0, and 0 < p < ∞. Then the function log Mp,α(zk, r) is convex in log r.
Proposition 5.
Proposition 5. Suppose k ≥0, −3 ≤α ≤0, and p = 2. Then the function log M2,α(zk, r) is convex in log r.
Proposition 5. Suppose k ≥0, −3 ≤α ≤0, and p = 2. Then the function log M2,α(zk, r) is convex in log r.
Proposition 6.
Proposition 6. Suppose α ̸∈[−3, 0] and p = 2. Then there exist positive integers k such that the function log M2,α(zk, r) is not convex in…
Proposition 6. Suppose α ̸∈[−3, 0] and p = 2. Then there exist positive integers k such that the function log M2,α(zk, r) is not convex in log r for r ∈(0, 1).
Proposition 7.
Proposition 7. Suppose hk(x) is a sequence of positive and twice differ- entiable functions on (0, 1) such that the function H(x) = ∞ X k=0…
Proposition 7. Suppose {hk(x)} is a sequence of positive and twice differ- entiable functions on (0, 1) such that the function H(x) = ∞ X k=0 hk(x) is also twice differentiable on (0, 1). If for each k the function log hk(x) is convex in log x, then log H(x) is also convex in log x.
Theorem 8.
Theorem 8. Suppose f ∈H(D) and −3 ≤α ≤0. Then the function r 7→log M2,α(f, r) is convex in log r. Moreover, the range −3 ≤α ≤0 is best…
Theorem 8. Suppose f ∈H(D) and −3 ≤α ≤0. Then the function r 7→log M2,α(f, r) is convex in log r. Moreover, the range −3 ≤α ≤0 is best possible.
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