Reducing inequality isn't just good for democracy, it's good for the economy as a whole.
That's the conclusions of the International Monetary Fund, the World Bank, and other capitalist outlets.
Let's go straight to their findings at the IMF.

Earlier IMF work has shown that income inequality matters for growth and its sustainability. Our analysis suggests that the income distribution itself matters for growth as well. Specifically, if the income share of the top 20 percent (the rich) increases, then GDP growth actually declines over the medium term, suggesting that the benefits do not trickle down. In contrast, an increase in the income share of the bottom 20 percent (the poor) is associated with higher GDP growth. The poor and the middle class matter the most for growth via a number of interrelated economic, social, and political channels.

The study then goes on to say that "income inequality matters for growth, we show that the income distribution itself matters for growth as well." No this isn't the Daily Worker, but the home of capitalist thought.
Why is this true? The study explains, "studies have argued that a prolonged period of higher inequality in advanced economies was associated with the global financial crisis by intensifying leverage, overextension of credit, and a relaxation in mortgage-underwriting standards (Rajan 2010), and allowing lobbyists to push for financial deregulation (Acemoglu 2011)." Furthermore, "enhanced power by the elite could result in a more limited provision of public goods that boost productivity and growth, and which disproportionately benefit the poor (Putnam 2000; Bourguignon and Dessus 2009)."
Notice the tie-in between wealth inequality and power over government policies.
This study is 39 pages long and full of charts in case you were unconvinced.

Let's go back to that "earlier IMF work" from 2011.

We find that longer growth spells are robustly associated with more equality in the income distribution. For example, closing, say, half the inequality gap between Latin America and emerging Asia would, according to our central estimates, more than double the expected duration of a growth spell.

So it isn't just the rate of growth that less inequality would enhance, but the duration of that economic growth as well. This makes sense when you consider the study above which associates greater wealth inequality with overextensions of credit, a common feature in speculative bubbles.

These findings aren't limited to the IMF. A 2014 study at the Organisation for Economic Co-operation and Development (OECD) found similar results.

Reducing income inequality would boost economic growth, according to new OECD analysis. This work finds that countries where income inequality is decreasing grow faster than those with rising inequality.

The single biggest impact on growth is the widening gap between the lower middle class and poor households compared to the rest of society. Education is the key: a lack of investment in education by the poor is the main factor behind inequality hurting growth.

“This compelling evidence proves that addressing high and growing inequality is critical to promote strong and sustained growth and needs to be at the centre of the policy debate,” said OECD Secretary-General Angel Gurría. “Countries that promote equal opportunity for all from an early age are those that will grow and prosper.”

Rising inequality is estimated to have knocked more than 10 percentage points off growth in Mexico and New Zealand over the past two decades up to the Great Recession. In Italy, the United Kingdom and the United States, the cumulative growth rate would have been six to nine percentage points higher had income disparities not widened, but also in Sweden, Finland and Norway, although from low levels. On the other hand, greater equality helped increase GDP per capita in Spain, France and Ireland prior to the crisis.

The OECD, like the IMF, is not anything remotely close to a socialist outlet. The OECD then turned around and released another study 9 months later that reached the same conclusion.

The World Bank looked at the issue from a slightly different perspective, but drew the same conclusion.

We find that high overall inequality only appears to hurt income growth of the poor. While inequality is also found to have a positive effect on growth, this positive effect is exclusively reserved for the top end of the income distribution. This means that the type of growth that inequality stimulates is the type that further advances inequality.

The study goes on to say that "high levels of top in equality serve as a proxy for “social separatism” where the rich lobby for policies that benefit themselves but ultimately limit the growth opportunities of the poor."

For generations, conservatives, libertarians, and free-market capitalists have been telling us that inequality is a good thing, because more rewards for the rich means that they will be encouraged to work harder which causes more economic activity.
It sounds good in theory, but we now see the truth to be entirely different.

Scientific evidence shows that more inequality leads only to more financial speculation, which causes harmful economic bubbles, and then leads the wealthy to buy off corrupt politicians to change to the rules so they can protect their wealth from natural readjustments.

Progressive should note this. For generations, progressives have been arguing against inequality on the basis of fairness, and what is good for society.
While those things are true, conservatives and capitalists couldn't care less about them.
Instead, what progressives should do is stress that reducing inequality is good for general economic growth. That way you can directly undermine the argument against redistribution.