WEIGHTED INTEGRALS OF ANALYTIC FUNCTIONS ARISTOMENIS G. SISKAKIS This paper is dedicated to Professor Ferenc M´ oricz on his 60th birthday. Abstract. We derive a formula of a weight v in terms of a given weight w such that the estimate  D |f (z)| p w(z) dm(z) ∼|f (0)| p +  D |f  (z)| p v(z) dm(z), is valid for all analytic functions f on the unit disc. 1. Introdunction Let D be the unit disc in the complex plane C and dm(z)= rdr dθ π the normalized Lebesgue area measure on D. Our starting point is the estimate  D |f (z)| p dm(z) ∼|f (0)| p +  D |f  (z)| p (1 -|z|) p dm(z), which is valid when 1 ≤ p< ∞ for all analytic functions on the disc. The notation means that there are finite positive constants C and C  independent of f (but possibly depending on p) such that the left and right hand sides L(f ) and R(f ) satisfy CR(f ) ≤ L(f ) ≤ C  R(f ) for all analytic f . In particular the two sides are either both infinite or both finite and in the latter case they are comparable. The weighting factor (1 -|z|) p in the second integral compensates for the extra growth of the derivative as z approaches the boundary. It is well known that a similar formula holds when the integrals are taken with respect to more general measures dμ(z) = (1 -|z|) α dm(z), α> -1:  D |f (z)| p (1 -|z|) α dm(z) ∼|f (0)| p +  D |f  (z)| p (1 -|z|) p+α dm(z), and one can find in the literature other cases of analogous estimates. For example it was shown in Proposition 5 of [AS] that if w(r), 0 <r< 1, is a positive weight function which is integrable on (0, 1) and satisfies the conditions: (1.1) w(r) ≥ C 1 - r  1 r w(u) du, for 0 <r< 1. for some positive constant C, and (1.2) w(sr +1 - s) ≥ C  w(r), for 0 <r< 1, 1991 Mathematics Subject Classification. 46E15, 30E99. This article was finished while the author was visiting Purdue University in Spring 1998. The author thanks the Mathematics Department for the hospitality. 1