arXiv:2610.40889v1 Announce Type: new Abstract: We study the class $CC_\alpha$ of completely convex functions of order $\alpha$, defined through a two-point analogous of the classical convexity condition. We give several equivalent characterizations of $CC_\alpha$, together with representation formulas in terms of two auxiliary Schwarz functions linked by a compatibility equation. From these representations, we derive coefficient bounds, a subordination result $zf'/f$, and a two-point Taylor expansion in which the Schwarzian derivative $S_f$ arises naturally as the coefficient of the linear term. This yields a pointwise bound for $S_f$ and a bound for the pre-Schwarzian norm $\|P_f\|_{CC_\alpha}$, improving the classical Yamashita bound when $\alpha\leq 1-1/\sqrt2 $. We also show that, unlike the class $C_\alpha$ of convex functions of order $\alpha$, every $f\in CC_\alpha$ is bounded when $\alpha>0$.
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