In the present paper, we will study geometric properties of harmonic mappings whose analytic and co-analytic parts are (shifted) generated functions of completely monotone sequences.

We call the solution of a kind of second order homogeneous partial differential equation as real kernel α − \alpha - harmonic mappings. For this class of mappings, we explore its Heinz type inequality. Furthermore, for a subclass of real kernel α − \alpha - harmonic mappings with real coefficients, we estimate their coefficients. At last, we study the extremal function of Schwartz type lemma for the class of real kernel α − \alpha - harmonic mappings.

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In [26], Olofsson introduced a kind of second order homogeneous partial differential equation. We call the solution of this equation real kernel ?-harmonic mappings. In this paper, we study some geometric properties of this real kernel ?-harmonic mappings. We give univalence criteria and sufficient coefficient conditions for real kernel ?-harmonic mappings that are fully starlike or fully convex of order ?, ? ? [0, 1). Furthermore, we establish a Landau type theorem for real kernel ?-harmonic mappings.

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