Sharing economy services like AirBnB, Lyft, Uber, et al reduce income inequality as follows:
Once upon a time, only rich people who could afford to build hotels could offer rooms for guests to rent. Thanks to AirBnB, now anyone with a home or apartment can offer a room for rent. Hence, income inequality reduced.
Once upon a time, only rich people who could afford to buy a taxi fleet or medallion could offer car rides to other passengers. Thanks to Lyft and Uber, now anyone with a car can offer rides to other passengers. Hence, income inequality reduced.
The same dynamic of reduced income inequality will play out in each new sharing category. Delivery, field service, personal care, etc. This expansion of economic opportunity and reduction of inequality is a direct result of widespread deployment of disruptive tech, the smart phone. As history has repeatedly shown, putting means of production and technology in the hands of the masses increases their opportunity and income.
Since we are at the very beginning of understanding the full range of real world applications of the new wave of technology, we are also at the very start of the creation of thousands of new opportunities for economic growth and progress for a huge number of people.