Kompleks düzlemde Bernstein-Walsh eşitsizlikleri
Bernstein – Walsh inequalities in complex plane
Yazar: CEVAHİR DOĞANAY GÜN
Danışman: PROF. DR. FAHREDDİN ABDULLAYEV
Mersin Üniversitesi / Fen Bilimleri Enstitüsü / Matematik Ana Bilim Dalı
Derece: Doktora, 2014
Konu: Matematik = Mathematics

Özet:
\[\mathbb{C}\] kompleks düzlem, \[G\subset \mathbb{C}\],  \[L:=\partial G\] Jordan eğrisi ile sınırlı sonlu bir bölge ve \[\Omega :=\overline{\mathbb{C}}\backslash \overline{G}\] olsun. \[w=\Phi \left( z \right)\] ile \[\Omega \] bölgesinden \[\Delta :=\left\{ w:\left| w \right|>1 \right\}\] bölgesine tanımlı, \[\Phi \left( \infty  \right)=\infty \] ve \[\underset{z\to \infty }{\mathop{\lim }}\,\frac{\Phi \left( z \right)}{z}>0\] 
 koşullarını sağlayan konform ve yalınkat dönüşüm gösterilsin. \[h\left( z \right)\], \[{G}'\supseteq G\]'de tanımlı ağırlık fonksiyonu olsun. Her \[p>0\] için${{A}_{p}}\left( h,G \right)$ ile \[G\] bölgesinde analitik ve
	\[{{\left\| f \right\|}_{{{A}_{p}}\left( h,G \right)}}={{\left( \iint\limits_{G}{h\left( z \right){{\left| f\left( z \right) \right|}^{p}}d{{\sigma }_{z}}} \right)}^{\frac{1}{p}}}<\infty \] 
koşulunu sağlayan fonksiyonlar sınıfı; \[L\] kapalı, ölçülebilir eğri olmak üzere, ${{\mathcal{L}}_{p}}(h,L)$ ile \[L\] üzerinde integrallenebilen ve
	\[{{\left\| f \right\|}_{{{\mathcal{L}}_{p}}(h,L)}}:={{\left( \int\limits_{L}{h\left( z \right){{\left| f(z) \right|}^{p}}\left| dz \right|} \right)}^{\frac{1}{p}}}<\infty \]
koşulunu sağlayan fonksiyonlar sınıfı gösterilsin. 
Her $n\in \mathbb{N}$ için ${{\wp }_{n}}$ ile derecesi $n$'yi aşmayan cebirsel polinomların sınıfı gösterilsin. Bu tez'de; $X:=C(\overline{G})$; $Y$:= ${{\mathcal{L}}_{p}}\left( h,\partial G \right)$ veya $Y:={{A}_{p}}\left( h,G \right)$ olmak üzere, her ${{P}_{n}}\in {{\wp }_{n}}$ için aşağıdaki iki problem kompleks düzlemin çeşitli bölgelerinde incelenmiştir: 

	\[\begin{align}
  & 1.{{\left\| {{P}_{n}} \right\|}_{X}}\le {{\mu }_{n}}\left( h,G \right){{\left\| {{P}_{n}} \right\|}_{Y}},{{\mu }_{n}}(h,G)\to \infty ,\,\,\,n\to \infty , \\ 
 & 2.\left| {{P}_{n}}(z) \right|\le \frac{{{\eta }_{n}}(h,G)}{d(z,L)}\,\,{{\left\| {{P}_{n}} \right\|}_{Y}}{{\left| \Phi (z) \right|}^{n+1}},\,\,\,\,\,z\in \Omega ,{{\eta }_{n}}(h,G)\to \infty ,\,\,\,n\to \infty . \\ 
\end{align}\] 
Yani, \[{{\mu }_{n}}\] ve \[{{\eta }_{n}}\] sayılarının \[\infty \]'a gitmesi, \[h\] ve \[G\]'nin geometrik özelliklerine bağlı olarak incelenmiştir.

Abstract:
Let \[\mathbb{C}\]  be complex plane,  \[G\subset \mathbb{C}\]  be a finite region bounded by a Jordan curve \[L:=\partial G\]  and \[\Omega :=\overline{\mathbb{C}}\backslash \overline{G}\] . Let \[w=\Phi \left( z \right)\] be the univalent conformal mapping of \[\Omega \]  onto the \[\Delta :=\left\{ w:\left| w \right|>1 \right\}\]  normalized by \[\Phi \left( \infty  \right)=\infty \]  and \[\underset{z\to \infty }{\mathop{\lim }}\,\frac{\Phi \left( z \right)}{z}>0\] . Let \[h\left( z \right)\]  be a weight function defined in \[{G}'\supseteq G\] . For any \[p>0\] , let us  denote by ${{A}_{p}}\left( h,G \right)$  the class of functions f  which are analytic in \[G\]  and satisfying the condition
	\[{{\left\| f \right\|}_{{{A}_{p}}\left( h,G \right)}}={{\left( \iint\limits_{G}{h\left( z \right){{\left| f\left( z \right) \right|}^{p}}d{{\sigma }_{z}}} \right)}^{\frac{1}{p}}}<\infty \]   
Let ${{\mathcal{L}}_{p}}(h,L)$  denote the class of functions f which are integrable and satisfying the condition
	 ,\[{{\left\| f \right\|}_{{{\mathcal{L}}_{p}}(h,L)}}:={{\left( \int\limits_{L}{h\left( z \right){{\left| f(z) \right|}^{p}}\left| dz \right|} \right)}^{\frac{1}{p}}}<\infty \]
when \[L\]  is rectifiable curve.
Let ${{\wp }_{n}}$  denote the class of arbitrary algebraic polynomials of degree at most $n\in \mathbb{N}$ . In this thesis, for any ${{P}_{n}}\in {{\wp }_{n}}$, following two problems have been investigated in various regions of the complex plane:
\[\begin{align}
  & 1.{{\left\| {{P}_{n}} \right\|}_{X}}\le {{\mu }_{n}}\left( h,G \right){{\left\| {{P}_{n}} \right\|}_{Y}},{{\mu }_{n}}(h,G)\to \infty ,\,\,\,n\to \infty , \\ 
 & 2.\left| {{P}_{n}}(z) \right|\le \frac{{{\eta }_{n}}(h,G)}{d(z,L)}\,\,{{\left\| {{P}_{n}} \right\|}_{Y}}{{\left| \Phi (z) \right|}^{n+1}},\,\,\,\,\,z\in \Omega ,{{\eta }_{n}}(h,G)\to \infty ,\,\,\,n\to \infty . \\ 
\end{align}\] 

 
where  $X:=C(\overline{G})$; $Y$:= ${{\mathcal{L}}_{p}}\left( h,\partial G \right)$, $Y:={{A}_{p}}\left( h,G \right)$.
It is shown that \[{{\mu }_{n}}\] and \[{{\eta }_{n}}\]  tends to \[\infty \]  depending on geometric properties of  \[h\]  and  \[G\].