Bohr-type inequalities for harmonic mappings using multiple Schwarz functions

Abstract This paper investigates Bohr-type inequalities for K -quasiconformal harmonic mappings in the unit disk <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi mathvariant="double-struck">D</m:mi> <m:mo>≔</m:mo> <m:mo>{</m:mo> <m:mi>z</m:mi> <m:mo>∈</m:mo> <m:mi mathvariant="double-struck">C</m:mi> <m:mspace width="0.17em"/> <m:mo>:</m:mo> <m:mspace width="0.17em"/> <m:mo>|</m:mo> <m:mi>z</m:mi> <m:mo>|</m:mo> <m:mo>&lt;</m:mo> <m:mn>1</m:mn> <m:mo>}</m:mo> </m:math> $\mathbb{D} := \left\{z\in \mathbb{C}\,:\,\vert z\vert &lt; 1\right\}$ by incorporating multiple Schwarz functions into the majorant series. We establish several improved and refined versions of the Bohr inequality that generalize and interconnect numerous known results. Our approach not only systematically recovers the existing theorems as special cases but also generates new results that are inaccessible through single-function methods.