We consider the class of all sense‐preserving harmonic mappings f=h+g¯ of the unit disk double-struckD, where h and g are analytic with g(0)=0, and determine the Bohr radius if any one of the following conditions holds: 1.h is bounded in double-struckD. 2.h satisfies the condition Re h(z)≤1 in double-struckD with h(0)>0. 3.both h and g are bounded in double-struckD. 4.h is bounded and g′false(0false)=0. We also consider the problem of determining the Bohr radius when the supremum of the modulus of the dilatation of f in double-struckD is strictly less than 1. In addition, we determine the Bohr radius for the space scriptB of analytic Bloch functions and the space BH of harmonic Bloch functions. The paper concludes with two conjectures.

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We consider the class of all sense-preserving harmonic mappings f = h + g of the unit disk D, where h and g are analytic with g(0) = 0, and determine the Bohr radius if any one of the following conditions holds:(1) h is bounded in D.(2) h satisfies the condition Re h(z) ≤ 1 in D with h(0) > 0.(3) both h and g are bounded in D.(4) h is bounded and g ′ (0) = 0. We also consider the problem of determining the Bohr radius when the supremum of the modulus of the dilatation of f in D is strictly less than 1. In addition, we determine the Bohr radius for the space B of analytic Bloch functions and the space B H of harmonic Bloch functions. The paper concludes with two conjectures.