Apple shares rose on Tuesday, further narrowing their decline over the past year, after an investment bank issued a bullish note on iPhone demand.

Demand for the smartphone, which is Apple’s most important product, is tracking at 56.5m units for the quarter to the end of March, according to an analysis of web search data by Morgan Stanley. That is “clearly ahead of expectations as investors are sceptical Apple can reach implied guidance of 52m,” the New York-based investment bank said.

Anxiety has been rising about slowing sales growth for the iPhone, which accounted for more than two-thirds of Apple’s total revenue in the previous quarter, due to saturation in the domestic market and economic headwinds from China, a key growth driver for the world’s biggest publicly traded company.