1. The problem is to compare and understand the values of the three numbers: the mixed number $1 \frac{1}{3}$, the decimal $1.4$, and the fraction $\frac{127}{100}$.\n\n2. Convert the mixed number $1 \frac{1}{3}$ to an improper fraction:\n$$1 \frac{1}{3} = \frac{3 \times 1 + 1}{3} = \frac{4}{3}$$\n\n3. Convert the improper fraction $\frac{4}{3}$ to a decimal to compare easily:\n$$\frac{4}{3} = 1.333\ldots$$\n\n4. The decimal $1.4$ is already in decimal form.\n\n5. Convert the fraction $\frac{127}{100}$ to decimal:\n$$\frac{127}{100} = 1.27$$\n\n6. Now compare the decimals:\n- $1 \frac{1}{3} = 1.333\ldots$\n- $1.4$\n- $\frac{127}{100} = 1.27$\n\n7. Order from smallest to largest:\n$$1.27 < 1.333\ldots < 1.4$$\n\nFinal answer: The values in increasing order are $\frac{127}{100}$, $1 \frac{1}{3}$, and $1.4$.