On some summability methods and the Tauberian role of Ramaswami \(\phi, \psi\) in the theory of entire and meromorphic functions (Q1970267)

The core of the article goes back to results developed by the first author [Proc. Am. Math. Soc. 14, 497-500 (1963; Zbl 0122.31703)] the growth of a positive increasing function \(f\) and \[ f^*(x)= \int^x_1 f(t)t^{-1} dt, \] in particular their type with respect to \(x^\rho\), given by lim sup or lim inf of \(f*(x) x^{-\rho}\) as \(x\to\infty\) and analogously for \(f^*\). The statement of these results involves the Ramaswami functions, given in \([0,1]\) by \[ (1-\phi(t))\exp \phi(t)= t\quad\text{and}\quad (1+ \psi(t))\exp(-\psi(t))= t, \] and many of these results are optimal. The original application was to relate the counting function \(n(x)\) of the zeros of an entire function \(f\) to its growth. The authors describe cases where similar principles apply. There is an extensive bibliography.