Let ψ be a conformal map on D with ψ(0)=0 and let F_α={z∈D:|ψ(z)|=α} for α>0. Denote by H^p(D) the classical Hardy space with exponent p>0 and by h(ψ) the Hardy number of ψ. Consider the limits L:=lim_{α→+∞} (log ω_D(0,Fα)^{−1} / logα), μ:=lim_{α→+∞}(d_D(0,Fα) / logα), where ω_D(0,Fα) denotes the harmonic measure at 0 of F_α and d_D(0,Fα) denotes the hyperbolic distance between 0 and F_α in D. We study a problem posed by P. Poggi-Corradini. What is the relation between L, μ and h(ψ)? Motivated by the result of Kim and Sugawa that h(ψ)=lim inf_{α→+∞} (log ω_D(0,Fα)^{−1} logα), we show that h(ψ)=lim inf_{α→+∞} (d_D(0,Fα) / logα). We also provide conditions for the existence of L and μ and for the equalities L=μ=h(ψ). Poggi-Corradini proved that ψ∉H^μ(D) for a wide class of conformal maps ψ. We present an example of ψ such that ψ∈H^μ(D).

This is the first of a sequence of papers proving the quantum invariance under ordinary flops over an arbitrary smooth base. In this first part, we determine the defect of the cup product under the canonical correspondence and show that it is corrected by the small quantum product attached to the extremal ray. We then perform various reductions to reduce the problem to the local models. In Part II, we develop a quantum Leray--Hirsch theorem and use it to show that the big quantum cohomology ring is invariant under analytic continuations in the Kähler moduli space for ordinary flops of splitting type. In Part III, together with F. Qu, we remove the splitting condition by developing a quantum splitting principle, and hence solve the problem completely.

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