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Lipschitz Properties of Harmonic and Holomorphic Functions
Ravisankar, Sivaguru

2011, Doctor of Philosophy, Ohio State University, Mathematics.

We prove two results in this dissertation, one concerning Lipschitz harmonic functions and the other concerning Lipschitz holomorphic functions.


Let B be a regular majorant. We show that a harmonic function, in a smoothly bounded domain Ω in ℝn, that is Lipschitz-B along a family of curves transversal to bΩ is Lipschitz-B in Ω (i.e., Lipschitz-B in all directions in Ω).


Let Ω be a smoothly bounded domain in ℂn (n > 1). Let PbΩ and let νP be the outward unit normal to bΩ at P. Fix PbΩ and a unit vector v⃗ ∈ ℂn. For δ > 0, we define R(Pδ ; v⃗), where Pδ = P−δνP, to be the radius of a complex disc centred at Pδ in the v⃗ direction that fits inside Ω̅ satisfying some additional properties. We show that a Lipschitz-B holomorphic function in Ω has a Lipschitz gain along complex discs centred at Pδ in the v⃗ direction. This gain is given by the inverse of R(Pt ; v⃗) as function of t. Some examples including an application to convex domains of finite type in ℂn are discussed.

Jeffery McNeal (Advisor)
Kenneth Koenig (Committee Member)
Saleh Tanveer (Committee Member)
59 p.

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Ravisankar, S. (2011). Lipschitz Properties of Harmonic and Holomorphic Functions. (Electronic Thesis or Dissertation). Retrieved from https://etd.ohiolink.edu/

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Ravisankar, Sivaguru. "Lipschitz Properties of Harmonic and Holomorphic Functions." Electronic Thesis or Dissertation. Ohio State University, 2011. OhioLINK Electronic Theses and Dissertations Center. 25 Sep 2017.

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Ravisankar, Sivaguru "Lipschitz Properties of Harmonic and Holomorphic Functions." Electronic Thesis or Dissertation. Ohio State University, 2011. https://etd.ohiolink.edu/

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