Precise fundamental inequalities and sum of deficiencies (Q1177974)

In 1959 W. K. Hayman proved the following theorem: Let \(f\) be a transcendental meromorphic function in the plane and \(k\) be a positive integer. Then \[ T(r,f)<(2+1/k)N\left(r,{1\over f}\right)+(2+2/k)\overline N\left(r,{1\over f^{(k)}-1}\right)+S(r,f). \] The author improves this result remarkably. Under the same assumptions he shows that for \(\varepsilon>0\) \[ T(r,f)<(1+1/k)N\left(r,{1\over f}\right)+(1+1/k)N\left(r,{1\over f^{(k)}-1}\right)-N\left(r,{1\over f^{(k+1)}}\right)+\varepsilon T(r,f)+S(r,f). \] He also proves several inequalities for meromorphic functions and their derivatives and gives some estimates to the sum of deficiencies.