David Cameron has told an audience in Mumbai he will make it easier for businessmen and students to come to Britain - but in return he wants India to tear down barriers to investment from the UK.
He also launched another broadside against aggressive tax avoidance , warning that in return for low tax rates, businesses must be prepared to pay their fair share.
Mr Cameron is keen to drum up business for British firms in India, where he is leading the largest trade mission ever to travel overseas with a prime minister.
On the first day of his visit, the PM has also laid a wreath for police victims of the 2008 Mumbai terror attacks and found time to tweet a picture of himself playing cricket with local youngsters.
But there is a risk that his trip will be clouded by corruption allegations surrounding the sale of luxury AgustaWestland helicopters to India.
Mr Cameron announced the creation of a new "same-day" visa service for Indian businessmen and said there was "no limit" to the number of Indians allowed to study at UK universities and stay on in graduate-level jobs.
Speaking to workers at the Mumbai headquarters of the Anglo-Dutch Unilever group, Mr Cameron said: "Britain is one of the most open, easy-to-invest-in countries in the world. We are incredibly welcoming.
"I think, in return, we should be having a conversation about opening up the Indian economy, making it easier to do business here, allowing insurance and banking companies to do more foreign direct investment into the Indian economy."
He added: "We are looking with your government at whether we can open up a whole corridor between Mumbai and Bangalore of growing towns and developments, and work and plan that with you."
Mr Cameron is accompanied on his three-day trip by a delegation of more than 100 representatives of major corporations, small businesses and academic institutions, as well as football's Premier League, the London Underground and nine parliamentarians.
He said he may raise the possibility of India buying part British-made Eurofighter Typhoon jets over the French preferred bidder for a £6.4bn contract when he meets the country's prime minister and president tomorrow.
On tax, the PM warned: "I believe in low taxes. Governments should be trying to get their rates of tax down so they are competitive, but then I think it is only fair to ask businesses to pay them.
"The message to business should be, 'If we are cutting this rate of tax down to a good low level, you should be paying that rate of tax, rather than seeking ever more aggressive ways to avoid it'."
Mr Cameron also said the Conservative Party, British business and the judiciary need to do more to get women into senior positions.
He revealed that his wife, Samantha, has pressed him for action on the issue.
He said: "My wife likes to say that if you don't have women in the top places, you are not just missing out on 50% of the talent, you are missing out on a lot more than 50% of the talent - and I think she probably has a point."
The Prime Minister offered a message of support to British employees of helicopter manufacturer AgustaWestland, which has been mired in corruption allegations.
He said it was up to the Serious Fraud Office to decide whether it should investigate after the chief executives of AgustaWestland and its parent company were arrested, sparking fears that a £480m deal with India could be affected.
But he told workers at the company's factory in Yeovil, Somerset: "What I am telling people here is that AgustaWestland is an excellent company, with highly-skilled workers who make brilliant helicopters."
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