### Abstract

We prove the inequality h(x)^{−1} G(x, y)h(y) < cG(x, y) +c, where G is the Green function of a plane domain D, h is positive and harmonic on D, and c is a constant whose value depends on the topological nature of the domain. In particular, for the class of proper simply connected domains c may be taken to be an absolute constant. As an application, we prove the Conditional Gauge Theorem for plane domains of finite area for which the constant c in the above inequality is finite.

Original language | English (US) |
---|---|

Pages (from-to) | 721-733 |

Number of pages | 13 |

Journal | Transactions of the American Mathematical Society |

Volume | 318 |

Issue number | 2 |

DOIs | |

State | Published - 1990 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

**A conformal inequality related to the conditional gauge theorem.** / McConnell, Terry R.

Research output: Contribution to journal › Article

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TY - JOUR

T1 - A conformal inequality related to the conditional gauge theorem

AU - McConnell, Terry R

PY - 1990

Y1 - 1990

N2 - We prove the inequality h(x)−1 G(x, y)h(y) < cG(x, y) +c, where G is the Green function of a plane domain D, h is positive and harmonic on D, and c is a constant whose value depends on the topological nature of the domain. In particular, for the class of proper simply connected domains c may be taken to be an absolute constant. As an application, we prove the Conditional Gauge Theorem for plane domains of finite area for which the constant c in the above inequality is finite.

AB - We prove the inequality h(x)−1 G(x, y)h(y) < cG(x, y) +c, where G is the Green function of a plane domain D, h is positive and harmonic on D, and c is a constant whose value depends on the topological nature of the domain. In particular, for the class of proper simply connected domains c may be taken to be an absolute constant. As an application, we prove the Conditional Gauge Theorem for plane domains of finite area for which the constant c in the above inequality is finite.

UR - http://www.scopus.com/inward/record.url?scp=0041579895&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0041579895&partnerID=8YFLogxK

U2 - 10.1090/S0002-9947-1990-0957083-8

DO - 10.1090/S0002-9947-1990-0957083-8

M3 - Article

AN - SCOPUS:0041579895

VL - 318

SP - 721

EP - 733

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 2

ER -