In this article we give an expository account of the holomorphic motion theorem based on work of Mãne-Sad-Sullivan, Bers-Royden, and Chirka. After proving this theorem, we show that tangent vectors to holomorphic motions have |ǫ log ǫ| moduli of continuity and then show how this type of continuity for tangent vectors can be combined with Schwarz’s lemma and integration over the holomorphic vari...

We show local H\"older continuity of quasiminimizers functionals with non-standard (Musielak--Orlicz) growth. Compared previous results, we cover more general minimizing and need fewer assumptions. prove Harnack's inequality a Morrey type estimate for quasiminimizers. Combining this Ekeland's variational principle, obtain $\omega$-minimizers.

In order to estimate the solution of zero for nonlinear systems, we conduct local convergence investigation in this paper. contrast Lipschitz condition used preceding study, have Hölder continuity requirement. Additionally, use a derivative approximation take free iterative technique with same order. A computed radius balls based on constant is also provided. No Taylor's series higher Fréchet d...