Sharp Hardy constants for annuli

We consider the Hardy inequality in canonical doubly connected plane domains. For any annulus A we determine sharp Hardy's constant c2(A) in function of conformal modulus M(A). Namely, for any annulus A with fixed conformal modulus M(A)=M we prove thatc2(A)={1/4,if M∈(0,M⁎];γ(2−γ)/4,if M∈(M⁎,∞), where γ=γ(M)∈(1,2). The critical modulus M⁎≈0.57298 and the values of γ(M) are found as roots of certain equations, containing the Gauss hypergeometric functions. In particular, we show that the sharp Hardy constants c2(A) depend on M continuously and that they tend to zero as M→∞. In addition, we describe an application of results to a Rellich type inequality.