Bohr phenomenon for certain subclasses of harmonic mappings

The Bohr phenomenon for analytic functions of the form f(z) = Sigma(infinity)(n=0) a(n)z(n), first introduced by Harald Bohr in 1914, deals with finding the largest radius r(f), 0 < r(f) < 1, such that the inequality Sigma(infinity)(n=0) vertical bar a(n)z(n)vertical bar <= 1 holds whenever the inequality vertical bar f(z)vertical bar <= 1 holds in the unit disk D = {z is an element of C : vertical bar z vertical bar < 1}. The exact value of this largest radius known as Bohr radius, which has been established to be r(f) = 1/3. The Bohr phenomenon [1] for harmonic functions f of the form f(z) = h(z) + <(g(z))over bar>, where h(z) = Sigma(infinity)(n=0) a(n)z(n) and g(z) = Sigma(infinity)(n=1) b(n)z(n) is to find the largest radius r(f), 0 < r(f) < 1 'such that Sigma(infinity)(n=1) (vertical bar a(n)vertical bar + vertical bar b(n)vertical bar vertical bar z vertical bar(n) <= d(f(0), partial derivative f(D)) holds for vertical bar z vertical bar <= r(f), here d(f(0), partial derivative f(D)) denotes the Euclidean distance between f(0) and the boundary of f(D). In this paper, we investigate the Bohr radius for several classes of harmonic functions in the unit disk D. (C) 2021 Elsevier Masson SAS. All rights reserved.