On Certain Bounds of Harmonic Univalent Functions - Fethiye Müge Sakar; Omendra Mishra; Georgia Irina Oros; Basem Aref Frasin (Axioms)
Harmonic functions are renowned for their application in the analysis of minimal surfaces. These functions are also very important in applied mathematics. Any harmonic function in the open unit disk <semantics>U=<mfenced separators="" open="{" close="}">z∈C:<mfenced open="|" close="|">z</mfenced><1</mfenced></semantics> can be written as a sum <semantics>f=h+<mover>g¯</mover></semantics>, where h and g are analytic functions in <semantics>U</semantics> and are called the analytic part and the co-analytic part of f, respectively. In this paper, the harmonic shear <semantics>f=h+<mover>g¯</mover>∈SH</semantics> and its rotation <semantics>fμ</semantics> by <semantics>μ<mspace width="4pt"></mspace><mfenced separators="" open="(" close=")">μ∈C,<mfenced open="|" close="|">μ</mfenced>=1</mfenced></semantics> are considered. Bounds are established for this rotation <semantics>fμ</semantics>, specific inequalities that define the Jacobian of <semantics>fμ</semantics> are obtained, and the integral representation is determined.