  Constructing theorems can help to determine the shape of certain utility
curves that make up the new definitions in financial markets. The aim of this
study was to present proofs for these theorems. Basic thoughts of new
alternative definitions emerge from the decision-making under uncertainty in
economics and finance. Shape of the certain utility curve is central to
standard definitions in determining risk attitudes of investors. Shape alone
determines risk behavior of investors in standard theory. Although the terms
risk-averse, risk-loving, and risk-neutral are equivalent to strict concavity,
strict convexity, and linearity, respectively, in standard theory, strict
concavity or strict convexity, or linearity are valid for certain new
definitions, not being the same as standard theory. Hence, it can be stated
that new alternative definitions are broader than standard definitions from the
viewpoint of shape. For instance, the certain utility curve of a risk-averse
investor can be strictly concave or strictly convex, or linear in alternative
definitions.
