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Let H be the class of complex-valued harmonic mappings f = h + g ¯ defined in the unit disk D , where h and g are analytic functions in D with h (0) = 0 = h ′ (0) - 1 and g (0) = 0 and H 0 = { f = h + g ¯ ∈ H : g ′ (0) = 0 } and P H 0 (M) : = { f = h + g ¯ ∈ H 0 : Re (z h ″ (z)) > - M + | z g ″ (z) | , z ∈ D , M > 0 } . Functions in the class P H 0 (M) are known to be fully starlike univalent functions for 0 < M < 1 / log 4 . In this paper, we obtain the sharp Bohr–Rogosinski type inequality and improved Bohr inequality and the corresponding Bohr radius for the class P H 0 (M) . [ABSTRACT FROM AUTHOR]

In this article, we consider the family of functions f analytic in the unit disk | z | < 1 with the normalization f (0) = 0 = f ′ (0) - 1 and satisfying the condition | (z / f (z) ) 2 f ′ (z) - 1 | < λ for some 0 < λ ≤ 1 . We denote this class by U (λ) and we are interested in the relations between U (λ) and other families of functions holomorphic or harmonic in the unit disk. Our first example in this direction is the family of functions convex in one direction. Then, we are concerned with the subordinates to the function 1 / ((1 - z) (1 - λ z)) . We prove that not all functions f(z)/z (f ∈ U (λ)) belong to this family. This disproves an assertion from [14]. Further, we disprove a related coefficient conjecture for U (λ) . We consider the intersection of the class of the above subordinates and U (λ) concerning the boundary behaviour of its functions. At last, with the help of functions from U (λ) , we construct functions harmonic and close-to-convex in the unit disk. [ABSTRACT FROM AUTHOR]

Recent investigations into what geometric properties are preserved under the convolution of two planar harmonic mappings on the open unit disk $${\mathbb {D}}$$ have typically involved half-plane and strip mappings. These results rely on having a convolution that is locally univalent and sense-preserving on $${\mathbb {D}}$$ , and thus, much focus has been on trying to satisfy this condition. We introduce a family of right half-strip harmonic mappings, $$\Psi _c : {\mathbb {D}}\rightarrow {\mathbb {C}}$$ , $$c>0$$ , and consider the convolution $$\Psi _c * f$$ for a harmonic mapping $$f = h +\overline{g}: {\mathbb {D}}\rightarrow {\mathbb {C}}$$ . We prove it is sufficient for $$h \pm g$$ to be starlike for $$\Psi _c *f$$ to be locally univalent and sense-preserving. Moreover, $$\Psi _c * f$$ decomposes into a convex combination of two harmonic mappings, one of which is f itself. This decomposition is key in addressing mapping properties of the convolution, and from it, we produce a family of convex octagonal harmonic mappings as well some other families of convex harmonic mappings. Additionally, motivated by the construction of $$\Psi _c$$ , we introduce a generalized harmonic Bernardi integral operator. We demonstrate convolution preserving properties and a weak subordination relationship for this extended operator. [ABSTRACT FROM AUTHOR]