{
  "statement": "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.\n\nProve H5: For every \\(n\\ge1\\), every \\(A\\in M_n(\\mathbb C)\\), and every \\(f\\) holomorphic on a neighborhood of\n\\[\nW(A)=\\{x^*Ax:x\\in\\mathbb C^n,\\ \\|x\\|_2=1\\},\n\\]\none has\n\\[\n\\|f(A)\\|\\le2\\max_{z\\in W(A)}|f(z)|.\n\\]\nThe norm is the Euclidean operator norm, \\(f(A)\\) is defined by holomorphic functional calculus, and the constant \\(2\\) is optimal uniformly over \\(n,A,f\\).",
  "starting_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.",
  "target_statement": "For every \\(n\\ge1\\), every \\(A\\in M_n(\\mathbb C)\\), and every \\(f\\) holomorphic on a neighborhood of\n\\[\nW(A)=\\{x^*Ax:x\\in\\mathbb C^n,\\ \\|x\\|_2=1\\},\n\\]\none has\n\\[\n\\|f(A)\\|\\le2\\max_{z\\in W(A)}|f(z)|.\n\\]\nThe norm is the Euclidean operator norm, \\(f(A)\\) is defined by holomorphic functional calculus, and the constant \\(2\\) is optimal uniformly over \\(n,A,f\\).",
  "case_id": "OA-1-H4-to-H5",
  "given_context": {
    "premise": "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."
  },
  "protocol": "single"
}