Linear Differential Equations and Logarithmic Derivative Estimates

Abstract

We prove two sharp inequalities for the growth of solutions of certain linear differential equations in the unit disk. For the proofs of these inequalities, we use the method of successive approximations and sharp estimates for the logarithmic derivatives of finite order meromorphic functions in the unit disk. These techniques can also be used to give an alternate proof of a well‐known result in the plane. The sharp logarithmic derivative estimates are a corollary of general estimates, and all these estimates have independent interest. 2000 Mathematics Subject Classification 34M10, 30D35.

Linear differential equations with solutions in the growth space H_\omega^\infty

“…For slowly growing solutions some other methods may give better results. Some useful techniques are, for example, Gronwall's lemma [7], Herold's comparison theorem [11], Picard's successive approximations [2,5] and methods based on Carleson measures [10,13,14,15]. Moreover, in the case of the complex plane, Wiman-Valiron theory is a commonly used method [12].…”

Sharp Logarithmic Derivative Estimates With Applications to Ordinary Differential Equations in the Unit Disc

“…The findings concerning (1.5) are summarized in the following theorem, which partially improves the corresponding results in [2,3,13]. Note that (1.7) below is proved in [13], but it has been included here for the sake of completeness.…”

“…Sharp growth estimates for the maximum modulus of the generalized logarithmic derivative f (k) / f ( j) , where f is meromorphic in D and k and j are integers satisfying k > j ≥ 0, are obtained in [3,8]. The special cases where f either belongs to the Nevanlinna class or is a Blaschke product are further discussed in [9].…”

“…For a measurable set E ⊂ [0, 1), the upper density is defined as D(E) = lim sup [3] Sharp logarithmic derivative estimates 147…”

Properties of higher order differential polynomials generated by solutions of complex differential equations in the unit disc

“…We need to give some definitions and discussions. Firstly, let us give two definitions about the degree of small growth order of functions in ∆ as polynomials on the complex plane C. There are many types of definitions of small growth order of functions in ∆ (see [10] , [11]) . If b < ∞, we say that f is of finite b degree (or is non-admissible).…”