GAUSSIAN INTEGRAL MEANS OF ENTIRE FUNCTIONS: LOGARITHMIC CONVEXITY AND CONCAVITY

Theorem 1. Let $\alpha < 0$ . Suppose both $x \mapsto M(x)$ and $x \mapsto M'(x)$ are positive on $(0, \infty)$ . Then the function $$x \mapsto \ln \frac{\int_0^x M(t)e^{-\alpha t}dt}{\int_0^x e^{-\alpha t}dt}$$ is convex in $\ln x$ for x in an open interval $I \subset (0, \infty)$ provided that the following conditions are satisfied: (i) $x \mapsto \ln M(x)$ is convex in $\ln x$ for $x \in I$ ; ²⁰¹⁰ Mathematics Subject Classification. Primary 30H10, 30H20. Key words and phrases. logarithmic convexity, logarthmic concavity, Gaussian integral mean, entire function. Jie Xiao is supported in part by NSERC of Canada and URP of Memorial University. (ii) $$x\frac{M'(x)}{M(x)} \ge \frac{[x(1-\alpha x)-\varphi(x)][\alpha\varphi(x)^2-2(1+\alpha x)\varphi(x)+2x]}{(\varphi(x)-x)[x-(1+\alpha x)\varphi(x)]} \text{ for } x \in I.$$ As a straightforward consequence of Theorem 1, the following logarithmic convexity for $M_{p,\alpha}(f,\cdot)$ is similar to Corollary 8 in [2].

Corollary 2. Let $(\alpha, p) \in (-\infty, 0) \times (0, \infty)$ . If $f : \mathbb{C} \to \mathbb{C}$ is an entire function, then $r \mapsto \ln \mathsf{M}_{p,\alpha}(f, r)$ is convex in $\ln r$ for $r \in \left(0, \sqrt{\frac{t_0}{-\alpha}}\right)$ , where $t_0 = 1.79 \cdots$ is the unique root of $u(t) = e^t - 1 - t - t^2$ on $(0, \infty)$ . During the process of extending Theorem 1 from I to $(0, \infty)$ , we find the following assertion.

Theorem 3. Let $\alpha < 0$ . Suppose both $x \mapsto M(x)$ and $x \mapsto M'(x)$ , are positive on $(0, \infty)$ . Then the function $$x \mapsto \ln \frac{\int_0^x M(t)e^{-\alpha t}dt}{\int_0^x e^{-\alpha t}dt}$$ is convex in $\ln x$ for $x \in (0, \infty)$ provided that $$\left(x\frac{M'(x)}{M(x)}\right)' \ge \begin{cases} 0, & x \in (0, x_0); \\ \frac{[x(1-\alpha x)-\varphi(x)]^2}{4x(\varphi(x)-x)^2}, & x \in [x_0, \infty), \end{cases}$$ where $x_0 = -t_0/\alpha$ is the unique root of $x(1 - \alpha x) - \varphi(x)$ on $(0, \infty)$ . As a by-product of Theorem 3, the following corollary extends the logarithmic convexity of $\mathsf{M}_{p,\alpha}(f,\cdot)$ from I to $(0,\infty)$ .

Corollary 4. Let $(\alpha, p) \in (-\infty, 0) \times (0, \infty)$ . If $f : \mathbb{C} \to \mathbb{C}$ is an entire function, then $r \mapsto \ln \mathsf{M}_{p,\alpha}(f, r)$ is convex in $\ln r$ for $r \in (0, \infty)$ if $$\left(x\frac{M'(x)}{M(x)}\right)' \ge \frac{\left[x(1-\alpha x) - \varphi(x)\right]^2}{4x[\varphi(x) - x]^2}, \qquad x \ge x_0,$$ where $x = r^2$ , $M(x) = \int_0^{2\pi} |f(\sqrt{x}e^{i\theta})|^p d\theta$ and $x_0$ is as the same as above. However, whenever handling the logarithmic concavity we have only one situation as follows.

Theorem 5. Let $\alpha \geq 0$ . Suppose M(x) & M'(x) are positive and M''(x) exists for $x \in (0, \infty)$ . If $x \mapsto \ln M(x)$ is concave in $\ln x$ for $x \in (0, \infty)$ , then the function $$x \mapsto \ln \frac{\int_0^x M(t)e^{-\alpha t}dt}{\int_0^x e^{-\alpha t}dt}$$ is also concave in $\ln x$ for $x \in (0, \infty)$ . Note that for any nonnegative integer k the classical integral mean of $z^k$ is both logarithmic convex and logarithmic concave. So, we obtain the following corollary, which is part (i) of [2, Theorem 7].

Corollary 6. Let $(\alpha, p) \in [0, \infty) \times (0, \infty)$ . If k is a nonnegative integer, then the function $r \mapsto \ln \mathsf{M}_{p,\alpha}(z^k, r)$ is concave in $\ln r$ for $r \in (0, \infty)$ . Notation. In the forthcoming sections, we will employ the symbol $\equiv$ when a new notation is introduced, but also use the notation $U \sim V$ when U and V have the same sign.

Lemma 7. Suppose f is positive and twice differentiable on $(0, \infty)$ . Then - (a) f(x) is convex in $\ln x$ if and only if $f(x^2)$ is convex in $\ln x$ and f(x) is concave in $\ln x$ if and only if $f(x^2)$ is concave in $\ln x$ . - (b) Let $$D(f(x)) \equiv \frac{f'(x)}{f(x)} + x \frac{f''(x)}{f(x)} - x \left(\frac{f'(x)}{f(x)}\right)^2.$$ Then $\ln f(x)$ is convex in $\ln x$ if and only if $D(f(x)) \ge 0$ and $\ln f(x)$ is concave in $\ln x$ if and only if $D(f(x)) \le 0$ for all $x \in (0, \infty)$ .

Lemma 8. Suppose $f = f_1/f_2$ is a quotient of two positive and twice differentiable functions on $(0, \infty)$ . Then $$D(f(x)) = D(f_1(x)) - D(f_2(x))$$ for $x \in (0, \infty)$ . Consequently, $\ln f(x)$ is convex in $\ln x$ if and only if $$D(f_1(x)) - D(f_2(x)) \ge 0$$ on $(0, \infty)$ and $\ln f(x)$ is concave in $\ln x$ if and only if $$D(f_1(x)) - D(f_2(x)) \le 0$$ on $(0, \infty)$ . We next establish several estimates for the function $\varphi$ .

Lemma 10. Given a nonconstant entire function $f: \mathbb{C} \to \mathbb{C}$ , suppose $$\begin{cases} \alpha \in \mathbb{R}; \\ x \in [0, \infty); \\ p \in (0, \infty); \\ M(x) \equiv M_p(f, \sqrt{x}) = \int_0^{2\pi} |f(\sqrt{x}e^{i\theta})|^p d\theta; \\ h = h(x) \equiv \int_0^r M_p(f, t)e^{-\alpha t^2} 2t dt = \int_0^x M(t)e^{-\alpha t} dt. \end{cases}$$ $$\begin{cases} A = A(x) \equiv \frac{\varphi(x) - x}{2}; \end{cases}$$ Let $$\begin{cases} A = A(x) \equiv \frac{\varphi(x) - x}{\varphi^2(x)}; \\ B = B(x) \equiv (1 - \alpha x) + x \frac{M'(x)}{M(x)}; \\ C = C(x) \equiv x \varphi'(x); \\ \Delta(x) \equiv D(h(x)) - D(\varphi(x)). \end{cases}$$ Then (a) $$S = S(x) = \sqrt{B^2 - 4AC} > 0 \ \forall \ x \in (0, \infty).$$ (b) $$\Delta(x) \sim -A \frac{h^2}{M^2} + B \frac{h}{M} - C \, \forall \, x \in (0, \infty)$$