Our goal is to analyse the inequality aspects of Human Development Index and to propose a new aggregation function that can adjust it by considering inequality penalisation. We take into account inequality across dimensions and across individuals and three laws inequality penalisation: decreasing, constant and increasing. At the beginning, we describe the features of standard Human Development Index and after we have surveyed main analytical contributions regarding the inequality adjustment of Human Development Index., Successively, we decline the basic properties of the human development indices and also we present specific properties enjoyed by the aggregation function proposed the Inequality Adjusted Exponential Mean (IAEM). This function is a specific case of the generalised mean. Three are the innovative aspects of IAEM function not enjoyed by the others ones. Firstly, the domain of IAEM function is unlimited. Secondly, IAEM function enjoys the property of incomplete compensability. Thirdly, with IAEM function it is possible to build three different rating and ranking classification according to the laws of penalisation. Finally, we apply the IAEM function to the database with 32 countries, developing and developed. According to the results the Inequality Adjusted Human Development Index built by the IAEM function is
significantly different from the standard Human development Index built by the arithmetic mean, expecially for the cases of decreasing and increasing penalisation. Moreover there is a negative correlation between the level of standard Human Development Index and the Penalisation Index, both in terms of rating and ranking.