A mathematical discussion on the relationship between price elasticity of demand and sales income

There exists obvious defects in the conclusions about the price elasticity of demand influence to sales income in many economics textbooks,because through strict mathematical analysis,the conclusion that lowering prices can increase sales income can be obtained in no way under the condition that the price elasticity of demand is larger than one.The correct conclusion should be this:in the same demand curve,there can be different elasticity to different points,and the price elasticity can be one in the middle of the curve,in which the sales income is also the largest,and the change in sales income can be decided by the position and moving direction of the sales place in demand curve.