Jack Dorsey founded payment start-up Square Inc. with a mission of rethinking the way people buy and sell stuff with their mobile phones.

Rising competition may get some merchants to rethink using Square.

Consider Fabrice Yopa, who used Square to collect payments for his online video business. Yopa ditched the service a month ago, saying he's fed up with how Square handled $4,000 in frozen subscriber payments.

"We've never had these kinds of problems with any other companies we use, including PayPal," Yopa said. "I would not recommend Square to anybody."

With Monday's debut of Apple Pay, a service that lets shoppers buy goods in stores using Apple Inc.'s iPhones, and the coming spinoff of eBay Inc.'s PayPal, more Square customers could be lured away by rival services that offer lower fees, easier transactions and a deeper pool of buyers and sellers.

The stiffening rivalry among Square, Apple, PayPal and other payment technology providers ­— all betting on a future when digital wallets replace cash and credit cards — reflects the rising stakes in a market that EMarketer Inc. estimates will be worth $118 billion by 2018, from $3.5 billion this year.

Square, co-founded by Twitter Inc. Chairman Dorsey, was one of the first companies to introduce technology that turns smartphones and tablets into credit-card readers. While Square appealed to smaller merchants seeking an easy way for customers to pay for goods and services, the start-up is seeking to expand into bigger retailers.

The key question is whether almost $600 million in funding over five years has given Square enough of an edge to fend off rivals, according to Scott Jacobson, managing director at Madrona Venture Group in Seattle.

"Square is going to have to figure out how to live or die on its own means," Jacobson said.

To do that, the company will have to keep customers like Leighton Morrison happy. Morrison said he signed up with Square to collect $3,000 in payments for a rental property, only to have the company freeze his funds, even after he sent documents to prove his identity and receipts of the transaction.

"I would never recommend them," Morrison said. "It's impossible to get someone on the phone."

Still, Square has signed up more than 1 million businesses since debuting in 2009, a person with knowledge of the data said. The San Francisco-based start-up processes about $30 billion in merchant sales annually and fees have generated about $900 million in revenue this year, according to the person, who asked not to be identified because the figures aren't public.

Square was valued at $6 billion in its latest round of financing this month, up from about $5 billion in January. The company has been the subject of reports that it has been in acquisition talks.

Executives regularly seek to quell deal speculation. Dorsey, Square's chief executive, denied a report by website Bank Innovation last week that PayPal was in early talks to acquire Square. After a 4,600-word report published by Fast Company in August questioned Square's viability, the company said in a blog post that it's well capitalized and spending money prudently.

"Reports that we tried to sell the company or of a delayed IPO? False," Square said in the post. "We're here for the long term."

So is PayPal. The firm, which will stay part of San Jose, Calif.-based eBay until later next year, is much bigger than Square and is furthest along in digital payments. It has 157 million active buyers and sellers completing transactions.

Another force to be reckoned with is Apple Pay, which rolled out Monday in the U.S. for the iPhone 6 and iPhone 6 Plus models. Apple is working with Stripe Inc., which enables payments in mobile applications, and has signed up banks and card issuers such as Visa Inc. and MasterCard Inc.

Rivals aren't sitting still either. Amazon.com Inc. introduced Amazon Local Register, a free application with a card reader that lets merchants accept payments. Google Inc. has a mobile wallet and has invested in services for merchants and shoppers.

"Just because you are Apple or Google, it doesn't mean you are going to win," said Gene Munster, an analyst at Piper Jaffray Cos. "But the stakes for Square are now much higher."

While Square, PayPal and Apple Pay will compete, they all approach the digital-payments market in different ways. Although it's too early to say which will come out on top in digital payments, consumers and merchants will be the ultimate winners, according to Todd Lutwak, former vice president of selling at eBay, who is now a partner at venture firm Andreessen Horowitz.

"Consumers and merchants are benefiting from the enormous competition that is going on, because payment fees are going down," Lutwak said in an interview.

Square takes 2.75 percent of every transaction and has a stamp-sized card reader, keeping a portion of that after paying processing fees to Visa, MasterCard and other card issuers that take on the task of protecting merchants and consumers against fraud. Whole Foods Market Inc. and Uniqlo Co. have started to use Square.

"Square really innovated the business model for the micro-merchant, but it doesn't have consumers," Karen Webster, CEO of Market Platform Dynamics in Boston, said in an interview. "You've got to get consumers to want a Square account, but how do you do that if there's nowhere to use it?"

Square made its first big merchant breakthrough in 2012 with Starbucks Corp., the largest coffee-shop chain, which invested $25 million in Square and adopted the company's technology in 7,000 stores. In 2013, the start-up introduced Square Stand, which uses an iPad to create a cash register and is marketed to merchants.