Duhamel Banach algebra structure of some space and related topics
Abstract: Let ? be a fixed complex number, and let ? be a simply connected region in complex plane C that is starlike with respect to ? ? ?. We define some Banach space of analytic functions on ? and prove that it is a Banach algebra with respect to the ?-Duhamel product defined by (f?? g)(z) := d/dz z?? f(z+??t)g(t)dt. We prove that its maximal ideal space consists of the homomorphism h? defined by h?(f)=f(?). Further, we characterize the lattice of invariant subspaces of the integration operator J?f(… Show more
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New norm inequalities for Jensen’s gap of analytic functions in Banach algebras with applications
Assume that h : G → ℂ {h:G\rightarrow\mathbb{C}} is analytic on the convex domain G and x ∈ ℒ ( ℬ ; E , 𝒜 , μ ) {x\in\mathcal{L}(\mathcal{B};E,\mathcal{A},\mu)} , the set of Bochner-integrable functions on a measurable space ( E , 𝒜 , μ ) {(E,\mathcal{A},\mu)} endowed with a countably-additive scalar measure μ on a σ-algebra 𝒜 {\mathcal{A}} of subsets of E and with values in the Banach algebra ℬ {\mathcal{B}} . If the spectrum σ ( x ( t ) ) ⊂ G {\sigma(x(t))\subset G} for all t ∈ E {t\in E} and γ ⊂ G {\gamma\subset G} is taken to be close rectifiable curve in G such that σ ( x ( t ) ) ⊂ ins ( γ ) {\sigma(x(t))\subset\operatorname*{ins}(\gamma)} for all t ∈ E {t\in E} , then, in this paper, we show among others that ∥ ∫ E ( h ∘ x ) ( u ) 𝑑 μ ( u ) - h ( ∫ E x ( u ) 𝑑 μ ( u ) ) ∥ ≤ 1 2 π ∫ E ∥ x ¯ E - x ( v ) ∥ ( ∫ γ | h ( ξ ) | ( | ξ | - ∥ x ¯ E ∥ ) ( | ξ | - ∥ x ( v ) ∥ ) | d ξ | ) 𝑑 μ ( v ) , \Bigg{\|}\int_{E}(h\circ x)(u)\,d\mu(u)-h\Bigg{(}\int_{E}x(u)\,d\mu(u)\Bigg{)}% \Bigg{\|}\leq\frac{1}{2\pi}\int_{E}\|\bar{x}_{E}-x(v)\|\Bigg{(}\int_{\gamma}% \frac{|h(\xi)|}{(|\xi|-\|\bar{x}_{E}\|)(|\xi|-\|x(v)\|)}|\,d\xi|\Bigg{)}\,d\mu% (v), where x ¯ E := ∫ E x ( u ) 𝑑 μ ( u ) . \bar{x}_{E}:=\int_{E}x(u)\,d\mu(u). Some examples for exponential function in Banach algebras are also given. Applications for discrete inequalities and Hermite–Hadamard-type inequalities are provided as well.
New norm inequalities for Jensen’s gap of analytic functions in Banach algebras with applications
Assume that h : G → ℂ {h:G\rightarrow\mathbb{C}} is analytic on the convex domain G and x ∈ ℒ ( ℬ ; E , 𝒜 , μ ) {x\in\mathcal{L}(\mathcal{B};E,\mathcal{A},\mu)} , the set of Bochner-integrable functions on a measurable space ( E , 𝒜 , μ ) {(E,\mathcal{A},\mu)} endowed with a countably-additive scalar measure μ on a σ-algebra 𝒜 {\mathcal{A}} of subsets of E and with values in the Banach algebra ℬ {\mathcal{B}} . If the spectrum σ ( x ( t ) ) ⊂ G {\sigma(x(t))\subset G} for all t ∈ E {t\in E} and γ ⊂ G {\gamma\subset G} is taken to be close rectifiable curve in G such that σ ( x ( t ) ) ⊂ ins ( γ ) {\sigma(x(t))\subset\operatorname*{ins}(\gamma)} for all t ∈ E {t\in E} , then, in this paper, we show among others that ∥ ∫ E ( h ∘ x ) ( u ) 𝑑 μ ( u ) - h ( ∫ E x ( u ) 𝑑 μ ( u ) ) ∥ ≤ 1 2 π ∫ E ∥ x ¯ E - x ( v ) ∥ ( ∫ γ | h ( ξ ) | ( | ξ | - ∥ x ¯ E ∥ ) ( | ξ | - ∥ x ( v ) ∥ ) | d ξ | ) 𝑑 μ ( v ) , \Bigg{\|}\int_{E}(h\circ x)(u)\,d\mu(u)-h\Bigg{(}\int_{E}x(u)\,d\mu(u)\Bigg{)}% \Bigg{\|}\leq\frac{1}{2\pi}\int_{E}\|\bar{x}_{E}-x(v)\|\Bigg{(}\int_{\gamma}% \frac{|h(\xi)|}{(|\xi|-\|\bar{x}_{E}\|)(|\xi|-\|x(v)\|)}|\,d\xi|\Bigg{)}\,d\mu% (v), where x ¯ E := ∫ E x ( u ) 𝑑 μ ( u ) . \bar{x}_{E}:=\int_{E}x(u)\,d\mu(u). Some examples for exponential function in Banach algebras are also given. Applications for discrete inequalities and Hermite–Hadamard-type inequalities are provided as well.
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