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<p>In this paper we propose a conjecture related to convex functions on the unit disc. Although the proposed result may initially seem technical and inapplicable, it ultimately reveals itself as a key component in a significant result concerning the preservation of convexity by the Graham–Kohr extension operator. The inequality put forward in the conjecture resolves the problem posed by the author regarding whether the Graham–Kohr extension operator preserves convexity. Thus, an extension problem in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathbb {C}^n\)</EquationSource> </InlineEquation> is reduced to an inequality involving convex functions in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathbb {C}\)</EquationSource> </InlineEquation>. We will approach the proposed conjecture from a numerical perspective, analyzing examples of convex functions on the unit disc. Although a rigorous proof is not yet available, this approach will nevertheless allow us to draw some conclusions regarding the validity of the conjecture.</p>

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A conjecture related to convex functions on the unit disc: numerical approach and extension results

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  • 发表日期：

    2026年4月20日

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  • Eduard Ştefan Grigoriciuc

摘要

In this paper we propose a conjecture related to convex functions on the unit disc. Although the proposed result may initially seem technical and inapplicable, it ultimately reveals itself as a key component in a significant result concerning the preservation of convexity by the Graham–Kohr extension operator. The inequality put forward in the conjecture resolves the problem posed by the author regarding whether the Graham–Kohr extension operator preserves convexity. Thus, an extension problem in \(\mathbb {C}^n\) is reduced to an inequality involving convex functions in \(\mathbb {C}\) . We will approach the proposed conjecture from a numerical perspective, analyzing examples of convex functions on the unit disc. Although a rigorous proof is not yet available, this approach will nevertheless allow us to draw some conclusions regarding the validity of the conjecture.