{
  "statement": "H3: Let \\(\\Omega\\subset\\mathbb C\\) be a bounded convex domain with \\(C^2\\) boundary, and let \\(\\gamma:[0,L]\\to\\partial\\Omega\\) traverse the boundary once counterclockwise at unit speed. Put \\(\\nu(s)=-\\mathrm i\\gamma'(s)\\).\nLet \\(B\\in M_n(\\mathbb C)\\) satisfy\n\\[\nW(B)=\\{x^*Bx:x\\in\\mathbb C^n,\\ \\|x\\|_2=1\\}\\subset\\Omega,\n\\]\nand define\n\\[\nR_s=(\\gamma(s)I-B)^{-1},\\qquad\nD_B(s)=\\frac1{2\\pi}\\left(\\nu(s)R_s+\\overline{\\nu(s)}R_s^*\\right).\n\\]\nThe map\n\\[\n\\Phi:C(\\partial\\Omega)\\to M_n(\\mathbb C),\\qquad\n\\Phi(h)=\\frac12\\int_0^L h(\\gamma(s))D_B(s)\\,ds\n\\]\nis positive and unital: \\(h\\ge0\\) implies \\(\\Phi(h)\\) is Hermitian positive semidefinite, and \\(\\Phi(1)=I\\).\n\nProve H4: For every \\(n\\ge1\\), every \\(B\\in M_n(\\mathbb C)\\) with pairwise distinct eigenvalues, and every \\(f\\) holomorphic on a neighborhood of\n\\[\nW(B)=\\{x^*Bx:x\\in\\mathbb C^n,\\ \\|x\\|_2=1\\},\n\\]\none has\n\\[\n\\|f(B)\\|\\le2\\max_{z\\in W(B)}|f(z)|.\n\\]\nThe norm is the Euclidean operator norm, and \\(f(B)\\) is defined by holomorphic functional calculus.",
  "starting_statement": "Let \\(\\Omega\\subset\\mathbb C\\) be a bounded convex domain with \\(C^2\\) boundary, and let \\(\\gamma:[0,L]\\to\\partial\\Omega\\) traverse the boundary once counterclockwise at unit speed. Put \\(\\nu(s)=-\\mathrm i\\gamma'(s)\\).\nLet \\(B\\in M_n(\\mathbb C)\\) satisfy\n\\[\nW(B)=\\{x^*Bx:x\\in\\mathbb C^n,\\ \\|x\\|_2=1\\}\\subset\\Omega,\n\\]\nand define\n\\[\nR_s=(\\gamma(s)I-B)^{-1},\\qquad\nD_B(s)=\\frac1{2\\pi}\\left(\\nu(s)R_s+\\overline{\\nu(s)}R_s^*\\right).\n\\]\nThe map\n\\[\n\\Phi:C(\\partial\\Omega)\\to M_n(\\mathbb C),\\qquad\n\\Phi(h)=\\frac12\\int_0^L h(\\gamma(s))D_B(s)\\,ds\n\\]\nis positive and unital: \\(h\\ge0\\) implies \\(\\Phi(h)\\) is Hermitian positive semidefinite, and \\(\\Phi(1)=I\\).",
  "target_statement": "For every \\(n\\ge1\\), every \\(B\\in M_n(\\mathbb C)\\) with pairwise distinct eigenvalues, and every \\(f\\) holomorphic on a neighborhood of\n\\[\nW(B)=\\{x^*Bx:x\\in\\mathbb C^n,\\ \\|x\\|_2=1\\},\n\\]\none has\n\\[\n\\|f(B)\\|\\le2\\max_{z\\in W(B)}|f(z)|.\n\\]\nThe norm is the Euclidean operator norm, and \\(f(B)\\) is defined by holomorphic functional calculus.",
  "case_id": "OA-1-H3-to-H4",
  "given_context": {
    "premise": "Let \\(\\Omega\\subset\\mathbb C\\) be a bounded convex domain with \\(C^2\\) boundary, and let \\(\\gamma:[0,L]\\to\\partial\\Omega\\) traverse the boundary once counterclockwise at unit speed. Put \\(\\nu(s)=-\\mathrm i\\gamma'(s)\\).\nLet \\(B\\in M_n(\\mathbb C)\\) satisfy\n\\[\nW(B)=\\{x^*Bx:x\\in\\mathbb C^n,\\ \\|x\\|_2=1\\}\\subset\\Omega,\n\\]\nand define\n\\[\nR_s=(\\gamma(s)I-B)^{-1},\\qquad\nD_B(s)=\\frac1{2\\pi}\\left(\\nu(s)R_s+\\overline{\\nu(s)}R_s^*\\right).\n\\]\nThe map\n\\[\n\\Phi:C(\\partial\\Omega)\\to M_n(\\mathbb C),\\qquad\n\\Phi(h)=\\frac12\\int_0^L h(\\gamma(s))D_B(s)\\,ds\n\\]\nis positive and unital: \\(h\\ge0\\) implies \\(\\Phi(h)\\) is Hermitian positive semidefinite, and \\(\\Phi(1)=I\\)."
  },
  "protocol": "single"
}