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The growth exponent of a function $f(\sigma + it)$ is defined as the least nonnegative real number $\psi(\sigma)$ satisfying $$ f(\sigma + it) \ll |t|^{\psi(\sigma) + \epsilon} $$ as $t \to \infty$, for each $\epsilon > 0$. The symbol $\ll$ should be read as the usual big-oh notation.

If $f(s, a)$ is a function of two complex variables $s, a$ analytic in s and a, respectively, and $\psi_{a}(\sigma)$ is defined as the least nonnegative real number satisfying $$ f(\sigma + it, a) \ll |t|^{\psi_{a}(\sigma) + \epsilon} $$ as $t \to \infty$, for each $\epsilon > 0$, then is there anything known about the continuity/convexity in the variable a, for fixed $\sigma$, of the function $\psi_{a}(\sigma)$ ?

Thanks.

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  • $\begingroup$ This is clearly not continuous, unless I am missing something. E.g. consider $f(s,a):= a\cdot s^2$, at $a=0$. $\endgroup$
    – Lasse Rempe
    Commented May 19, 2015 at 16:33
  • $\begingroup$ That's exactly right. Then, I tend to feel like asking if $\psi_{a}(\sigma) = 0$ for all $a$ in some interval $(c, d)$ implies $f(s, a) = 0$ for all $a$ in $(c, d)$, or not. But this question might seem to be too general. $\endgroup$
    – Mr. SnowRemover
    Commented May 20, 2015 at 10:46

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As mentioned in my comment, $\psi_a(\sigma)$ clearly does not depend continuously on $a$; consider $f(s,a) := a\cdot s^2$ at $a=0$.

For the question in your comment, note that this cannot remotely be true, even for the growth exponent in one variable. Indeed, e.g. by Arakelian's approximation theorem we can construct entire functions that are bounded on some arbitrary collection of vertical strips, while having arbitrarily fast growth on some other collection of vertical lines.

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  • $\begingroup$ My original question is resolved with Arakelian's approximation theorem; and yet, over the last several months, my interest has been in analyzing the growth exponent $\psi_{a}(\sigma)$ of integrals $\int_{0}^{1}f(\sigma + it, u)(u + a)^{-A}du$ with f being analytic in s and u in appropriate regions and growing at most polynomially in $t$ for each fixed $\sigma$, and A < 1. If you are generous enough, please give me your analysis on this problem too. Thanks. (I have been told of an example that for some nonanalytic f(s, u), $\psi_{a}(\sigma)$ could be made as big as one wish when $a = 0$.) $\endgroup$
    – Mr. SnowRemover
    Commented May 21, 2015 at 10:56
  • $\begingroup$ @Catman You might wish to post another question, giving all the details. (You could then post a link here if you wish.) $\endgroup$
    – Lasse Rempe
    Commented May 22, 2015 at 20:34

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