A refined cos\(\pi \rho\) theorem (Q2569913)

For an entire function \(f\in\mathcal O(\mathbb C)\), let \(M(r):=\max\{| f(z)| : | z| =r\}\), \(m(r):=\min\{| f(z)| : | z| =r\}\), and let \(n(r)\) denote the number of nonzero zeros of \(f\) in the disc \(\{| z| <r\}\). In the paper `\(\cos\pi\lambda\) again', Proc. Am. Math. Soc. 131, 1875--1880 (2003; Zbl 1015.30012), the author proved that if \(\liminf_{r\to+\infty}\log M(r)/r^\lambda=0\) for some \(\lambda\in(0,1)\), then \(\limsup_{r\to+\infty}\frac{\log m(r)-(\cos\pi\lambda)\log M(r)}{\log r}=+\infty\). Let \(\varPhi\) be an increasing unbounded function with \(\varPhi(r)=o(r^\rho)\) as \(r\to+\infty\). In the present paper the author shows that if \(f\) is of order \(\rho\) with \(\rho\in(0,1)\), \(\lim_{r\to+\infty}(n(r)-\alpha r^\rho)=-\infty\) with \(\alpha>0\), and \(| n(r)-\alpha r^\rho| \leq\varPhi(r)\), \(r\gg0\), then \(\limsup_{r\to+\infty}\frac{\log m(r)-(\cos\pi\rho)\log M(r)} {\varPhi(r)\log r}\geq-(1-\cos\pi\rho)\). The estimate is sharp -- the author constructs an entire function \(f\) with \(n(r)=r^\rho-\log r+O(1)\) such that \(\log m(r)-(\cos\pi\rho)\log M(r)\leq(-1/2)(1-\cos\pi\rho+o(1))\log^2r\).